Economic theory has neglected appropriation. A recognition of the peculiar and distinctive nature of appropriation (e.g., that it is not a return to a factor) has direct implications for some of the most controversial parts of economics such as capital theory. In recent decades, there has been the "Cambridge Controversy in Capital Theory" between the conventional neoclassical economists of Cambridge, Massachusetts and the neo-Keynesian, neo-Ricardian, and neo-Marxian economists of Cambridge, England. This controversy did clarify some matters, but it failed to shed much new light on the basic concepts of capital theory since both sides accepted the Fundamental Myth that the ownership of the product is attached to the ownership of the means of production. Some economists condoned and some condemned the "private ownership of the means of production." But without an appreciation of appropriation, neither side understood that the right to the product is not included in that ownership. The whole product is not pre-owned at all; it is appropriated.

This chapter sketches in a non-technical manner the implications of property appropriation for capital theory.

Broadly speaking, economic resources have two types of uses, "active" and passive.

A resource is used passively when it is sold or rented out in return for some market price or rental. A resource is used "actively" when, instead of being evaluated directly on the market, it is used up in production, usually along with other resources. Then the liabilities for the used-up resources and the rights to any produced assets are appropriated.

Appropriation is involved in the active use, not in the passive use of resources. Difficulties arise in the conventional treatment of the active case, since economic theory tends to ignore appropriation. The economic return in the active case is not just the value of the original resource but the extra value of the appropriated property. But the total return in the active case is mistakenly imputed only to the original resource, as if the ownership of the appropriated property were already included in ownership of the original resource. Property which is appropriated cannot be previously owned; otherwise it could not be appropriated. The extra value of the appropriated property (e.g., the whole product) is not a return to the original resource. In the context of the laissez faire appropriation mechanism, it is a return to the contractual role of being the hiring party, the last legal owner of the used-up resources.

Appropriation is neglected because the right to the whole product is treated as if it were part of a pre-existent property right. In the Marxist view of the capitalist economic system, the pre-existent property right is the "ownership of the means of production." In neoclassical capital theory, the pre-existent right is the ownership of a capital asset. Property appropriated in the future can have a present value (which could be zero) but it cannot have a present owner, since otherwise it could not be appropriated in the future. The primary imputation fallacy in capital theory is the "capitalized value" definition.

To illustrate fallacies involved in capital theory, we will use an extremely simple example. A capital good yields K units of capital services (machine-years) per year. Each year workers perform the labor L which uses up the capital services K and produces the output Q = f(K,L). The list or vector giving the positive and negative results of the year's production process is the production vector or

Whole Product = (Q, –K, –L).

Let P, R, and W be the unit prices of the outputs, capital services, and labor services respectively so the value of the whole product is the profit:

p = Profit = Market Value of Whole Product = PQ – RK – WL.

If r is the constant interest rate, then a future value FV at the end of one year has the present value PV = FV/(1+r). Suppose the capital good only yields K units of capital services for two years with no maintenance and then has no salvage value. Thus the net present value of the services yielded by the capital good is RK/(1+r) + RK/(1+r)2. The capital good has a current competitive price C. Arbitrage between the two possibilities of renting the capital good (buying the services K) or buying it will equalize the price of the good with the present value of the rental payments:

Market Cost = Capitalized Value of Rental Stream

One of the basic concepts of capital theory is the notion of the capitalized value of an asset. The definition is usually stated in a rather general fashion; owning the asset "yields" a future income stream and the discounted present value of the income stream is the capitalized value of the asset. But there are quite different ways in which "owning an asset" can "yield" an income stream. There are the "active" and the passive uses of capital. The capitalized value concept is unproblematic in the passive case where the income stream is the stream of rentals (net of maintenance) plus the scrap value. The capitalized value of that stream is, under competitive conditions, just the market cost C of the asset. Bonds and debentures provide similar examples of income streams generated by renting out or loaning out capital assets, i.e., by the passive use of capital.

Capital theory would be somewhat less controversial if it stuck to such examples of hired-out capital. However, the capitalized value definition is also applied to the quite different active case where, instead of hiring out the capital, labor is hired in, a product is produced and sold. In the example, the annuals net proceeds to the capital good owner acting as the employer are:

The present value of the stream of net proceeds is then called "the capitalized value V of the capital asset" as if to impute all the net proceeds to the capital asset:

Capitalized Value of the capital asset =

Thus the so-called "capitalized value V of the capital asset" is actually the market price of the capital asset C plus the present value ¸ of the property appropriated in the future. The rentals are the return to the capital asset; the property assets and liabilities underlying the profits are the whole products which are the return to the contractual role played by the capital owner (when the capital is used actively). The rights to the whole products are not part of the rights to the capital asset; whole products are appropriated.

The capitalized value definition overlooks appropriation. One might then think that by purchasing the asset or the "means of production," one is thereby purchasing the outputs and the net proceeds—so there is no need to appropriate the outputs.

This is incorrect. In fact one thereby purchases only the asset. Any further return will depend on one's contracts. If one rents out the asset and sells any scrap, then one receives only the rental-plus-scrap income stream. If, instead, one hires in labor, bears the costs of the used-up labor and capital services, and claims and sells the outputs, then one receives the net proceeds mentioned by Keynes. In each case, one owned the asset. The difference lies in the pattern of the subsequent contracts. By making the contracts so that one was the hiring party, one could additionally appropriate the whole product each time period with its positive or negative value. The capitalized value definition fallaciously imputes the value of the appropriated whole products to the capital assets rather than to the contractual role played by the capital owner.

Thus the problem with the "net productivity of capital" is that it fallaciously imputes the return to the capital-owner's social role (being the hiring party) to the capital asset itself.

Yet another method of imputing the whole product and its value, the profits, to capital is the "quasi-rent doctrine." In a genuinely competitive model, all factors including the services of fixed plant and equipment would have a competitively determined price. Capital assets would have a competitive rental. In conventional microeconomics, it is held that capital assets might "earn" a short run "quasi-rent" due to the short run inelasticities of supply in capital assets. But short run inelasticities do not require a special notion of "quasi-rents" in addition to the usual competitive rentals. Short term competitive rentals reflect such scarcities, and thus the short run rental might be higher than the rental in the longer term.

There is a pattern here. Capital has a passive use and an active use. Thus capital theory will always have a pair of concepts associated with capital, one concept derived for the passive case and one concept for the active case. The value concept associated with the active case includes the concept for the passive case plus the value of the whole product (the profits) which is appropriated by the capital owner in the active case.

The difference between the passive and active case is the appropriation of the whole product, which is the return to a contractual role, not a return to the capital. But conventional capital theory neglects appropriation, and imputes the whole product and its value, the profits, to capital.

One traditional reason for the neglect of the entire question of the appropriation of the whole product is that it doesn't matter, for the purposes of price theory, if attention is restricted to zero-profit perfectly competitive equilibrium.

The zero profit condition also occurs in capital theory where it hides the imputation of the whole product.

When the assignment of the whole products to the capital assets is challenged, the all-too-typical response is that it really doesn't matter because in perfectly competitive equilibrium the pure profits are zero. In that instance, Professor Samuelson can claim to have demonstrated the "EQUALITY OF CAPITALIZED VALUE AND REPRODUCTION COST" [Samuelson 1961, p. 42; reprinted in Samuelson 1966, p. 309], i.e., the equality of V and C in our example. Similarly, a prominent capital theorist shows that the competitive "equilibrium price of a one-year-old machine in terms of 'costs'" is equal to the "present discounted value of the future net output which a one-year-old machine can produce." [Burmeister 1974, p. 443]

In this special case, the fallacious imputation of the whole product to capital is a moot point from the value theoretic viewpoint since the whole products then have zero value. In property terms, the imputation is as incorrect in that case as in general. A property vector is not the zero vector even when its market value is zero. And, of course, the capital theoretic definitions of capitalized value or net productivity are by no means restricted to the competitive model in the economics and finance literature.

Professor Samuelson asserts that "capital goods have a 'net' productivity" [1976, p. 661] (while the other factors have only a marginal productivity), as a "technological fact" [1976, p. 600]. It is a clear-cut case where the social role of Capital as the hiring party in capitalist society is presented as a technological characteristic of capital goods.

Professor Samuelson's dictum that the cash value of a doctrine is in its vulgarization applies not only to Marxism but also to the fallacious imputations of neoclassical capital theory. For instance, in a book entitled The Capitalist Manifesto, one reads that:

The return to property is its rental or selling price. The product, i.e., the whole product, must be appropriated.

The recent controversies in capital theory did not get to the root of the matter because they remained at the value theoretic level. The root of the problem in capital theory is that it presents the return to Capital's market role (being the hiring party) as resulting from the technical and legal characteristics of capital goods. The whole product is presented as part of the "net productivity of capital" or "marginal efficiency of capital" and the legal rights to the whole product are presented as part of the "ownership of the means of production." In that regard, capital theory uses a high-brow version of the Fundamental Myth.

When the appropriation of the whole product is implicitly considered in the theory of the firm, the pattern, as in capital theory, is to construe the right to the whole product as being part of a pre-existent property right. In its commonest form, this property right is called the "ownership of the firm." We have defined the word "firm" to be the party who ends up appropriating the whole product:

"firm" = "whole product appropriator."

The identity of the firm (in this technical sense of whole product appropriator) is determined not by some pre-existent property right such as the so-called "ownership of the firm," but by who hires what or whom. Firmhood is a contractual role, not a property right.

Economists sometimes use a rather abstract version of the "ownership of a firm." Technical production possibilities are represented by a production function, a production set, or a "production-opportunity locus" [Hirshleifer 1970, p. 124], and then economists speak of the "owners" of these technical possibilities, e.g., the "owners of the productive opportunity" [Hirchleifer 1970, p. 125]. There is, however, no such ownership of a production function.

The answer to Haavelmo's "for what?" question is that the net profit is not the return to an already owned piece of property. The profits are the value of the whole product, the whole product is appropriated, and, via the market mechanism of appropriation, the whole product is appropriated by the last owner of the used-up inputs. Since economics overlooks appropriation, Haavelmo can only conceive of the profit income as a return to some existing property. Since the inputs have already been accounted for, he has to postulate the "ownership of a production function."

Neoclassical economics' lack of attention to property theoretic "details" is illustrated by the postulation of this peculiar "ownership" of a mathematical description of technically possible production opportunities such as a production function or production set. If one wishes to use the metaphor of appropriating the whole product as "trading with Nature," then one should realize that there are no "owners" [Hirchleifer 1970, p. 20] of the production set of possible trades with Nature. There might be the ownership of certain specialized inputs, such as proprietary technical information, but that is only the ownership of inputs to the production opportunity, not the "ownership" of the productive opportunity itself.

The notion of "ownership of a production set" is often intended as an abstract version of the ownership of a corporation. But, as we have seen, a corporation is an owner of certain inputs such as physical and financial capital. The legal process of incorporation does not magically convert the ownership of a capital asset into the ownership of the production set of net product vectors which could be produced using that capital asset.

For instance, if a person owns a widget grinder, that is the ownership of an input or factor of production. The person is a factor supplier, a supplier of a stream of capital services K of the widget grinder. Suppose the person then incorporates a corporation which issues shares to the person in return for the widget grinder. Then the person indirectly owns the widget grinder (i.e., owns the corporation which owns the grinder). Clearly this legalistic repackaging of the widget grinder ownership does not change anything in the argument about the non-ownership of firmhood. The identity of the firm is still determined by whether the owner of the capital asset (in this case, the corporation) hires out the asset or hires in labor.

Yet many economists suddenly consider the incorporated person as the owner of the set of production vectors which could be produced using the widget grinder. Instead of being considered a factor supplier, the person—now embodied as the corporation—has suddenly become a "firm"—a factor demander (demanding the complementary set of inputs to be used along with the widget grinder). The legal possibility of someone leasing the widget grinder from the corporation cannot even be represented in the economists' model of supply and demand schedules since the machine has been subsumed into the shape of the production set. Far from being so prosaic, economists have mystically interpreted the process of incorporation as a transubstantiation—the miraculous transformation of the ownership of a machine into the "ownership" of the production set of net output vectors that can be produced using the machine.

Often property theoretic reasoning about corporations is distorted by tautological formulations.

That is only a truism by the meaning of "its product." But what determines whether a given set of outputs produced using the corporation's machinery and buildings is "its product" or the product of some other party? That depends on the direction of the hiring contracts. If the corporation had leased out the plant and machinery (e.g., a widget grinder), the outputs would not be "its product." Alternatively, if the corporation had hired in a complementary set of inputs, then it would be in the legal position of the hiring party. By bearing the costs of production, it could lay legal claim to the outputs which thus become "its product." Hence truisms about a corporation owning "what it produces" or "its product" do not contradict the thesis that the appropriation of the outputs (and input liabilities) is determined by the direction of the hiring contracts.

In the early models of perfectly competitive equilibrium, constant returns to scale in production was assumed. This assured zero economic profits in equilibrium, so from the viewpoint of value theory, it was immaterial who was the firm, i.e., who appropriated the whole product vector (since it had zero net value).

In 1954, Professors Kenneth Arrow and Gerard Debreu published a paper [Arrow and Debreu 1954] in which they claimed to show the existence of a competitive equilibrium under the general conditions of non-increasing returns to scale, i.e., decreasing or constant returns to scale. Under decreasing returns to scale, there would be positive economic or pure profits. Hence the Arrow-Debreu model alleges to show the existence of a perfectly competitive equilibrium with positive economic profits. This result has now become part of the Received Truth in economics; Arrow and Debreu have each received the Nobel Prize in Economics.

The common metaphysical argument for universal constant returns to scale must be addressed before considering decreasing returns to scale. The argument is that if one doubles all the factors relevant to production, then the outputs will double. If not, then some relevant factor was not doubled. This argument, as stated, does not seem to be falsifiable. The argument is then applied to production functions in an economic model (quite a different matter). If doubling the inputs in a production function yields double the outputs, the function exhibits constant returns to scale. By Euler's Theorem in Calculus, there are zero profits in competitive equilibrium [see any microeconomics text].

The fallacy in the "universal constant returns" argument is the application of the general metaphysical argument to a production function in a competitive model. The general argument says "doubling everything doubles the outputs" but the application to the production function says "doubling the input variables doubles the outputs." The inputs in a production function do not represent everything that might be scarce and relevant to production. Indeed, in an economic model, the inputs have market prices associated with them so they represent exclusively owned marketable commodities that may be bought and sold on input markets. "Everything relevant to production" includes a myriad of other scarce factors affecting production such as: (1) commonly owned property or public goods (public roads, free parking, public parks, and such), and (2) unowned natural factors (air, river water, rainfall, sunlight, wind, oceans, and such).

The proposition that doubling just the exclusively owned inputs (the production function inputs) will double the outputs is a robustly empirical proposition and may well be false. The presence of scarce but not exclusively owned factors may introduce decreasing return to scale in the marketable inputs, and the latter is all that is required to have decreasing returns to scale in an economic model. Accordingly, Arrow and Debreu mention such a justification for decreasing returns to scale in their original article [see the mention of "free rationed goods" in Arrow and Debreu 1954, p. 267].

Since the whole product vectors can have a positive value in the Arrow-Debreu model, the model had to face the question as to how these vectors got assigned to people. McKenzie [1981], Koopmans [1957, p. 65], and others have interpreted the Arrow-Debreu model as assigning production sets to specific parties by postulating "hidden factors" owned by the parties. But this compromises the model in a number of ways [see Ellerman 1982, Chapter 13; or McKenzie 1981]. Firstly, there are no non-marketable privately owned input services, and Arrow and Debreu have identified none. The hidden factors which might justify decreasing returns are not privately owned (e.g., commonly-owned or unowned natural factors). The existence of unowned or commonly-owned factors does not account for the assignment of production sets to specific parties.

Professors Arrow and Hahn try to replace "not marketable" with "not marketed." But it is incoherent to simply assume that "not all inputs are, in fact, marketed" [Arrow and Hahn 1971, p. 61] when the production sets are first being specified.

Arrow and Hahn then restrict the whole product or production vectors to their "marketed" components, and leave the "private" components implicit in the shape of the production sets (all prior to the determination of any equilibrium prices). But whether or not an input is marketed or held for reservation uses will depend on the equilibrium configuration of prices (which are hardly known when production sets are first being specified).

The Arrow-Hahn tactic in not only methodologically incoherent; it could be inconsistent with the other assumptions. As Burmeister has pointed out:

Suppose an economic reform was instituted in Russia where some inputs were traded on free markets with factory managers instructed to maximize profits, but other inputs were designated as "not marketed" and were not exposed to market forces. Western economists would be quick to point out that if some factors were hidden from exposure to scarcity-reflecting market prices, then there could no assurance that an equilibrium would be allocatively efficient. Any "efficiency theorem" the Russians might derive would be bogus due to the existence of the non-marketed hidden factors that are not exposed to market signals. Unfortunately, economists seem to forget this critical but rather elementary insight when Arrow and Hahn use the same tactic and then claim to prove the "efficiency" of their model [p. 110].

What about fixed factors which are not marketed in order to reap the short-run quasi-rents over and above the long-run competitive rentals? Doesn't the ownership of these fixed factors determine the identity of firms, the assignment of production sets to legal parties? Physical plants and installed capital equipment are examples of fixed factors while direct labor and materials are variable factors. The time horizon needed to change plant size and install new capital equipment projects those changes into the intermediate or long run. In the interim, the owner of the fixed factor might earn a quasi-rent over the long-run competitive rental. Or at least that is the standard view.

This standard fixed factor argument contains a multitude of errors. "Free entry" in a competitive model does not require construction of new physical facilities. If there are economic profits then a higher rental can be offered for the existing facilities. The physical or technical fixity of factors does not make them legally fixed or legally immobile. Fixed factors can be and are leased in the short-run.

Hicks uses a version of the fixed factor argument based on resource specificity.

Hicks seems to be confusing the technical immobility which results from resource specificity with legal immobility. But specialized plant, equipment, and resources can be rented where they stand. Any model which does not allow that possibility does not model a competitive economy. Free entry does not require the long run construction of new physical facilities. Offer a higher rental for the existing facilities. The very same technical production process can be carried out under quite different legal auspices by rearranging the hiring contracts. Specialized factors are not "hidden" from such "takeover bids."

Returning to the general fixed factor argument, a comparison of short-run "quasi-rents" with long-run competitive rentals is a comparison of short-run apples and long-run oranges. The relevant point of comparison for short-run quasi-rents is with short-run competitive rentals which would reflect the short-run inelasticity of supply of fixed factors. As noted above, the efficient allocation of resources requires revealing the services of a fixed factor in the production set so those inputs will be exposed to the guidance of their market price, the short-run competitive rental. If the fixed factor was not hired out and the quasi-rent was below the short-run rental, the factor owner could do better by hiring it out. If the quasi-rent was the same as the short-run competitive rental, then we are back to the zero-profit model.

Thus we assume the quasi-rent is greater than the short-run competitive rental of the fixed factor. We may assume no other hidden privately-owned inputs; otherwise the above argument would just be repeated for them. Thus the difference between the quasi-rent and the competitive rental is the economic profit. Doesn't the higher quasi-rent on the fixed factor account for why the factor owner hires in a complementary set of inputs and undertakes production—instead of hiring out the factor at the competitive rental?

No—that strategic behavior would violate the assumption of competitive price-taking behavior by factor owners. Price-taking factor owners sell their factor to the market at the given price. A factor supplier can be both the seller and buyer of a commodity if both actions are consistent with price-taking behavior at the market price. The owner of the fixed factor has both the roles as the seller and buyer of the services of the fixed factor. The market price of those services is the given parametric short-run competitive rental. But once those services are offered on the market at the competitive rental, the alleged assignment of the production set to the fixed factor owner is lost. Any party could hire the fixed factor as well as the complementary inputs and reap the profits. Thus the competitive rental would be bid up; it could not be an equilibrium price.

Hence the fixed factor ploy fails, for several reasons, to account for the assignment of production sets to legal parties. If the factor services are not exposed to market forces, then the behavior of the factor owner may be inconsistent with the maximization assumptions. Once the factor is exposed to the market and the assumption of competitive price-taking behavior is enforced on the factor owner, the association of the production set with the factor owner is lost.

There is another interesting use of the fixed-factor ploy that might be mentioned. It is used to hide monopolistic power in a so-called "competitive" model. The conventional concept of monopoly power applied to input markets is like "justice" defined by the victors; it only applies to the vanquished. Conventional theory defines that a resource owner is monopolistic if the owner can affect the selling price of the resource, e.g., if the owner sells to a downward sloping demand curve. But it is only the losers in the hiring conflict who hire out their resource. The winner cannot be monopolistic in the sense of manipulating the selling price of his resource since he does not hire out his resource at all. He is the hiring party.

If one-hundred workers join together in a labor union to hire themselves out at a higher price, that is a combination in restraint of trade, a market imperfection, a monopolistic lump in the competitive soup. But if a hundred times as many capital owners pool their resources in a "capital union" called a joint stock company, then that is treated in conventional economics as a single producer. The combined capital owners are not being "monopolistic" because they have no designs to jack up the price of their capital services. Indeed, their purpose is not to hire out capital at all. The purpose of the capital union is to hire in labor and the complementary factors, to undertake production, and to sell the outputs. All the input and output markets actually represented in the conventional microeconomic model might be competitive.

Microeconomic models in the Marshallian tradition try to assign production functions to certain pre-selected individuals by awarding them a local monopoly on certain factors conveniently left implicit in the shape of the production functions. The model is still called "competitive" since the markets explicitly represented are competitive. The monopoly power of the specialized factor is used behind the scenes to win the hiring conflict and thus to completely eliminate the market in the monopolized factor. Such a model is hardly competitive, and it only appears so because the conventional notion of monopolistic market power is designed to apply only to the losers in the hiring conflict. In a genuinely competitive model, all resource owners supply their resources to the market at the going price—and thus resource ownership does not itself decide who is to be the firm.

Professor McKenzie interprets the Arrow-Debreu model as being based on indivisible hidden factors, but sees no reason to treat those factors differently from the marketed factors.

But this issue is not divisibility; it is marketability. Indivisible resources are rentable. Indivisible corporate resources, such as factories, can be and are leased out to other parties who could then exploit production opportunities using those resources as inputs. Thus indivisibility is irrelevant in the first place to the question of assigning production sets to legal parties. Moreover, if the competitive rental on the indivisible factor leaves no profit, then we are back in the zero-profit model.

The hidden factors ploy does not solve the problem of assigning production sets to economic agents. Also Arrow has explicitly ruled out hidden factors in the Arrow-Debreu model. In the following passage, Arrow contrasts the Arrow-Debreu model with a model by McKenzie [1959 or 1954] which used constant returns to scale.

Arrow explicitly states that "pure profits" are distributed to "the owners of the firm," and that, in contrast, the McKenzie model does not have this "residual category of owners who receive profits without rendering either capital or entrepreneurial services." Thus the Arrow-Debreu has a "residual category" of people who receive profits without supplying hidden inputs in the form of "capital or entrepreneurial services." They are the alleged "owners" of the production sets.

The key to assigning production sets to legal parties in the Arrow-Debreu model lies not in ad hoc and incoherent assumptions about non-marketed, fixed, or indivisible hidden factors. The key lies in the assumed structure of property rights, and that brings us back to the theme of property appropriation.

The Arrow-Debreu model assumes there are "owners of the firm," i.e., that there is a property right such as the "ownership" of the production sets of technically feasible whole product vectors. The train of reasoning is that production sets represent the production possibilities of "firms" and "firms" are identified with corporations which, of course, are owned by their shareholders.

The property theoretic error can be pin-pointed in the Arrow-Debreu model. Shareholders do indeed own corporations, but corporations do not own production sets. There is no problem in assuming that the ith consumer owns "a contractual claim to the share aij of the profit of the jth production unit [Arrow and Debreu 1954, p. 270] where "production unit" is a corporation. The problem comes in the assumption that for "each production unit j, there is a set Yj of possible production plans" [p. 267]. In a private enterprise capitalist economy, there is no such property right as the ownership of production sets of feasible whole product vectors.

In the Arrow-Debreu model each consumer-resourceholder is endowed prior to any market exchanges with a certain set of resources and with shares in corporations. But, prior to any market activity, ownership of corporate shares is only an indirect form of ownership of resources, the corporate resources. It is the subsequent contracts in input markets which will determine whether a corporation, like any other resource-owner, successfully exploits a production opportunity by purchasing the requisite inputs and appropriating the whole product.

The Arrow-Debreu model mistakes the whole logic of appropriation. The question of who appropriates the whole product of a production opportunity is not settled by the initial endowment of property rights. It is only settled in the markets for inputs by who hires what or whom. In other words, the determination of who is to be the "firm" (the whole product appropriator) is not exogenous to the marketplace; it is a market-endogenous determination. This adds a new degree of freedom to the model which can only be ignored in the special case of zero economic profits when it doesn't matter (for price theory) who is the firm. This new degree of freedom eliminates the possibility of a competitive equilibrium with positive economic profits, e.g., with decreasing returns to scale in some production opportunity.

There is no mathematical error in the Arrow-Debreu model. The model assumed a property right that in fact does not exist. By assuming that production possibilities are "owned," the Arrow-Debreu model does not allow anyone but the "owner" to demand the requisite inputs. Thus well-defined input demand schedules can be determined. But in a free enterprise capitalist economy, anyone can bid on the inputs necessary for some technically feasible production opportunity. In such an economy, production, the conversion of inputs into outputs, can be seen as a form of arbitrage, production arbitrage, between input markets and output markets.

Traditionally, arbitrage is thought of as an exchange operation, e.g., in currency markets. But if the price of Chicago wheat exceeds the price of Kansas City wheat plus the transportation costs, then the operation of buying inputs (Kansas City wheat plus transportation services) and selling the outputs (Chicago wheat) would still be called "arbitrage." If the price of a good one period in the future exceeds the current price plus storage costs, then

But in general equilibrium models such as the Arrow-Debreu model where commodities are differentiated by spatial and temporal location, transportation and storage are examples of production. As more characteristics of the inputs, besides spatial and temporal location, change in the production process, there is no magic dividing line which suddenly prevents the production arbitrage of buying all the requisite inputs and selling the outputs.

It is this concept of arbitrage applied to production itself, the concept of production arbitrage, which "kills" the Arrow-Debreu notion of a pure-profits equilibrium. When there is a sufficient price differential between input and output markets to allow positive profits, then potential production arbitrageurs (entrepreneurs) can attempt to reap those profits by purchasing the required inputs, bearing their costs as the inputs are consumed in production, claiming the produced outputs, and then selling the outputs. The demand for the inputs to profitable arbitrage opportunities in a frictionless economy can no more be modeled than can the demand for free money.

This critique of the Arrow-Debreu is the diametrical opposite of the usual criticisms that the idealized frictionless model does not fit the real world. The production arbitrage critique is particularly effective in the idealized model. Naturally, such a grand arbitrage operation would be difficult in the real world economy, but it is quite possible in the frictionless idealized model of perfect competition. There is no small irony here. The Arrow-Debreu "proof" of the existence of competitive equilibrium with positive economic profits is based on assuming a non-existent property right which rules out a certain form of arbitrage—a form of arbitrage that is perfectly possible in an idealized frictionless capitalist economy. Models that live by the sword of arbitrage must also be prepared to die by it.

How can the Arrow-Debreu model be repaired to obtain competitive equilibria with positive profits? It cannot be. A competitive equilibrium is not possible when there are profitable arbitrage opportunities, e.g., profitable production opportunities. Production arbitrageurs would bid up input prices. A competitive equilibrium is not possible in the situation which Arrow and Debreu attempt to model, a competitive capitalist economy with some production opportunities exhibiting decreasing returns to scale.

Some neo-classical economists are reconsidering the Arrow-Debreu model. Indeed, in reviewing a book about Nicholas Kaldor, Professor Frank Hahn [of Arrow and Hahn 1971] seems to be having second thoughts.

Citing modern work by McKenzie and others that does not assume the identity of firms to be given prior to market activity, Hahn concludes that Kaldor was "substantially right" [p. 1746].

This recognition restores a certain symmetry between increasing and decreasing returns to scale. A competitive equilibrium is not possible at a point of increasing returns to scale because no one wants to be the firm (negative profits). A competitive equilibrium is not possible at a point of decreasing returns to scale because everyone wants to be the firm (positive profits). General equilibrium for a competitive capitalist economy is only possible in the special case of constant returns to scale where (by assumption) no one cares who is the firm (zero profits).

Consider any proposed competitive equilibrium in a private enterprise capitalist economy where there are positive pure profits in some production opportunities. That economy can be modeled by a dollar-division game, and those profits are the "dollar." Since production sets are not "owned" and since the profits are a return to the contractual role of being the hiring party, any party forming a new coalition of resource suppliers can attempt to appropriate (the whole product whose value is) the profits, the "dollar." The proposed contracts in a profitable opportunity cannot represent an equilibrium because a potential production arbitrageur could use some of the profits to offer better terms to some of the resource suppliers.

A profitable competitive capitalist economy is a dollar-division game. All dollar-division games, as essential constant-sum games, have an empty core so there can be no competitive equilibrium with positive profits. The theorem that essential constant-sum games have an empty core can be restated as the theorem: if a constant-sum game has a non-empty core, then it is inessential. That is the game-theoretic argument that if a capitalist economy does have a competitive equilibrium, then it is with zero profits.

For the last several decades, the Arrow-Debreu model has been Received Truth in mathematical economics. Its collapse back to the constant returns case is a major example of the impact on economic theory of an appreciation of the nature and structure of property appropriation. The reason for its failure, which was uncovered by the analysis of appropriation, was the market-endogenous determination of "firmhood," of who is to appropriate the whole product and thus be the firm. Yet it is doubtful the guardians of Received Doctrine will openly admit that positive pure profits are incompatible with a genuine competitive equilibrium. In the short-run, Received Theory will continue to support the Arrow-Debreu model based on a hodge-podge of ad hoc and logically incoherent assumptions about hidden (non-marketed, fixed, or indivisible) factors or non-existent property rights to the ownership of production sets.

The extra degree of freedom, the market-endogenous determination of firmhood, cuts much deeper into Received Competitive Doctrine that just the Arrow-Debreu model. It changes the very conception of how competitive markets operate outside of the universal constant returns case—from an orderly process of equilibriation to a game-theoretically indeterminate struggle. In addition to the high-brow Arrow-Debreu model, this holds for the low-brow Marshallian partial "equilibrium" models used in current textbooks which show short-run pure profits.

The conventional theory is that there are two basic types of economic agents, consumer-resourceholders and "firms." The consumer/input-owners supply inputs to the input markets and demand outputs on the output markets. The firms play the opposite role of demanding inputs on input markets and supplying outputs to output markets. The flow of commodities from the consumers as input suppliers to the firms and the flow of products back to the consumers (with the money flows in the opposite direction) are represented in the familiar circular flow diagram.

The conventional picture assumes that firmhood is determined prior to market activity. The resource owners are lined up on one side and the "firms" are supposedly lined up on the other side of the input markets. But this is not the case in a free enterprise capitalist economy. It is not legally predetermined that an input owner is a supplier of inputs rather than a demander of a complementary set of inputs. In particular, it is not legally predetermined that a capital owner (corporate or not) is a labor demander rather than a capital supplier. Prior to the market contracts, corporations are just other input owners. Any resource owner, corporate or otherwise, may aspire to be a "firm" in the technical sense of "whole product appropriator" by attempting to purchase the complete set of inputs to a productive opportunity. Prior to market contracts, legal parties are not associated with production sets, so input demand and output supply schedules are not well-defined.

The customary analytical machinery of resource owners having input supply schedules and "firms" having input demand schedules prior to market activity incorrectly represents the structure of the market process. The identity of the firms (parties who will appropriate the whole products) is only determined at the end of the game-theoretically indeterminate market process, not at the beginning. Firmhood is also indeterminate in the special case of universal constant returns, but that indeterminacy does not affect income distribution (no profits) and at least the market (as opposed to firm) supply and demand curves are determinate in that case.

Under the impact of production arbitrage, the indeterminacy of firmhood leads to the breakdown of the conventional firm supply and demand schedules in competitive markets. And the possibility of production arbitrage was revealed by understanding the mechanism of property appropriation in a private property free enterprise economy.