To the editors:
In his essay about general equilibrium modeling, Philip Pilkington contends that there is no doubt about the existence of general equilibrium, but that stability has not been intuitively demonstrated. That is, it has not been convincingly shown that economic variables tend to follow paths of adjustment from disequilibrium values toward a set at which they will all be simultaneously in equilibrium. He bases these contentions on the history of general equilibrium theories, the improbability of general equilibrium being reached by chance, and a critique of current general equilibrium theory. This letter examines whether the methods and results found in representative examples of general equilibrium models support Pilkington’s views.
General Equilibrium Models
A major feature of both the history and present state of general equilibrium modeling is the distinction between virtual and non-virtual models. Another major feature is that they deal with purely competitive economies. Neither of these is discussed by Pilkington. He does not indicate explicitly the type of model to which his judgments refer, and he does not discuss important types of models in his critique.
General equilibrium modeling was first undertaken by Léon Walras in the late nineteenth century, and all subsequent general equilibrium modeling has its roots in his constructions.1 The models developed by Walras are largely concerned with economic tâtonnement, the equilibrating behavior of economic variables.2 In his first models, Walras dealt with a non-virtual, purely competitive system of markets in which variables are aggregated at the market level. In disequilibrium, commodities are bought irrevocably at prices at which market-desired supply and demand are not equal, and commodities are produced at prices that do not equal average costs. It would be tempting to call such phenomena real instead of non-virtual were it not for the fact that there are real markets that are also virtual: there are proposed transactions, but actual transactions occur only when the equilibrium price is quoted and at that price.3
In non-virtual models, who changes the price? The answer is of crucial importance for the stability of a model. Is there a mechanism whereby, in disequilibrium, prices and production are changed in the direction of an equilibrium set of values? Pilkington notes that Walras’s answer, describing real markets and his non-virtual model, is that prices are changed by the buyers and the sellers. When desired demand exceeds the amount buyers are able to purchase, “they raise the market price” and when desired supply exceeds the amount sellers are able to sell, “they lower the price.”4
In Walras’s model, there is also a tâtonnement of production:
If the average cost of production exceeds the price, output decreases; if the price exceeds the average cost of production, output increases; if the average cost of production is equal to the price, equilibrium is established.5
Obviously, as Walras wrote many times, the amount produced is changed by the “entrepreneur qui tâtonne.”6
In his non-virtual model, Walras believed that the existence of equilibrium is proved for the purely competitive economy on the grounds that the number of independent equations of supply and demand is equal to the number of unknowns, namely the prices and quantities. He believed that the stability of equilibrium is proved for his non-virtual model by the functioning of its pricing mechanism. Thought experiments with the consequences of changing prices, he contended, showed that
the system of new quantities produced and of their prices is nearer to equilibrium than its predecessor, and it is necessary only to continue the tatonnement in order to approach the equilibrium more and more closely.7
In actuality, he was making a reasonable, intuitive inference rather than providing a definitive proof, a point he acknowledged by stating that the stability of equilibrium in production, although not certain, is probable.8 Pilkington believes that Walras’s account of tâtonnement is not intuitively acceptable.
Equilibrium theory, and tâtonnement theory in particular, were not neglected following the death of Walras. The great economists who were his contemporaries and immediate successors enlarged upon the non-virtual version that he presented: Vilfredo Pareto developed a completely disaggregated model, as well as ideas about the entrepreneur and welfare economics; Knut Wicksell examined the integration of capital goods into the model; Enrico Barone contributed a neoclassical theory of distribution, a theory of public finance, and a general equilibrium model of a socialist economy; Irving Fisher added refinements to the theory of demand and of money; Henry Moore worked on a dynamic (moving) equilibrium and econometric general equilibrium model; Joseph Schumpeter modeled the features of economic cycles in relation to general equilibrium theory; many others also wrote on general equilibrium.9 All these economists accepted the proof of existence and account of the mechanisms of stability provided by Walras.
The non-virtual tâtonnement model posed a problem for Walras and his successors. Asset holdings are modeled as variables that enter into the structure and behavior of supply and demand functions. Changes of the array of individuals’ holdings—wealth, business inventories, personal possessions—change the individual and market-desired supply and demand quantities. Participants then make different economic choices, which cause the values—that is, the paths—of the variables to change. The equilibrium, if there is one, depends upon the paths, and therefore cannot be found or predicted by the use of the initial market desired supply and demand functions. These functions are no longer valid after disequilibrium transactions change the array. To predict the equilibrium, it would be necessary to specify the determinants of the paths of the variables. That would require knowing the series of multitudinous prices, transactions, changes of employment, wage rates, as well as productions, consumption, saving, and investment.
Walras and many other economists eliminated the theoretical difficulties arising from irrevocable disequilibrium transactions by assuming that they do not occur. One way this was achieved was by replacing the non-virtual model with a virtual one. Walras made a one-page sketch of such a model in 1899 and it was incorporated into the 1900 and 1926 editions of his treatise.10 In disequilibrium in the virtual tâtonnement, proposed prices are changed in the direction of the sign of the desired market excess demand. Production plans are changed in the direction of the proposed sign of price minus average cost, but no exchange or production, etc., takes place. There are therefore no changes during the tâtonnement in the array of asset holdings, and therefore no changes in the market supply and demand functions.
By what mechanism are prices and production changed in the direction of equilibrium in a virtual model? Pilkington notes that one such mechanism proposed by many theorists, and attributed by them to Walras, is a central auctioneer. That personage quotes prices, receives desired supply and demand information, and sets the market-clearing prices. Theoreticians have made the baseless supposition that Walras made that assumption explicitly or implicitly because, they suppose, buyers and sellers are exclusively price-takers in purely competitive markets, so there is no one to change the price unless it be a market authority.11 Pilkington observes that the recent translators of Walras, of whom I am one, have stated that it is an error to suppose that he employed such a personage. That supposition is indeed an error. Walras did not even mention an auctioneer. It is true that many years ago the young William Jaffé mistakenly translated “crier” as “auctioneer” on one occasion, even though a crier had functions that were quite different from those of an auctioneer. In addition, the crier was mentioned by Walras purely as part of a factual description of the Bourse. Jaffé can be forgiven his momentary lapse of concentration because he redeemed himself by describing the Walrasian auctioneer as a gratuitous invention by later economists; he added, “In fact, there is no ‘auctioneer’ anywhere in the Éléments. It is an invention after the fact made in later expositions and extrapolations of the original model.”12 Pilkington initially contends that the assumption of an auctioneer is not gratuitous, but is rather “an aide to thought.” He then decides that it is not really an aid; the complexity and enormity of the task of gathering, processing, and disseminating information about supplies and demands by a central auctioneer would be virtually impossible. He adds inexplicably that Walras was aware of that circumstance.13
Walras used the same invalid proof of existence for his virtual model as he had for the non-virtual one, doing so by implication because his virtual sketch is incomplete. He neglected to give demanders a role in the virtual tâtonnement. As for stability, Walras asserted that the price- and production-changing rules regarding excess desired quantities lead the virtual economy to the market-clearing general equilibrium set of prices and production given by the solutions to the initial market functions. When that state is reached, the model becomes non-virtual—that is, market-clearing trade, production, consumption, etc., take place. Neither Walras nor his successors in the virtual tradition proved that proposition. They relied on talking through a virtual process of equilibration to an alleged ending in general equilibrium, provided, Walras observed, that the parameters do not change.
Modeling in the 1930s
After 1930, a series of attempts were made to deal with the difficulties inherent in tracing the movement of variables in disequilibrium, difficulties that arise whether those variables are virtual or non-virtual. The models that are mentioned constitute the course of general equilibrium theorizing and help in evaluating Pilkington’s views.
One approach has been to eliminate tâtonnement altogether by assuming that the economy is always in equilibrium. Theorization was thereby vastly simplified because the problems of whether equilibrium exists and is unique, and whether the model is stable simply do not arise. There is no tâtonnement; the equilibrium is assumed to exist, not as a possibility but as the state of the model. That type of model was first developed by Karl Gustav Cassel in 1918.14 He was also the first economist to construct a model in which there is a moving general equilibrium—a path of growth for his model economy. Presumably, Pilkington would not endorse that type of model’s treatment of existence, since it is simply a postulate and, from a logical point of view, requires no proof.
Among other economists who assumed that the economy is always in equilibrium were Giulio La Volpe, John von Neumann, and John Hicks.15 In 1937, Neumann constructed a microeconomic dynamic growth model in which equilibrium exists and is unique.16 He achieved its results by making assumptions that ensure them. He assumed that the economy is purely competitive, that it is always in equilibrium, that every firm hires some of every input or produces some of every output, that capital goods are free, that land is free, and that consumption is determined exogenously by the minimum necessary for life. Investment is the excess of income over that amount. He deduced that the model economy is always in a moving equilibrium, growing at an endogenously determined rate. Obviously, an existence proof can be no more general than its assumptions. The assumptions, it can be seen, are extremely restrictive in Neumann’s case. Presumably, Pilkington would not use Neumann’s model as convincing evidence that existence is proven.
Hicks described his modeling as following in the tradition of “the Lausanne school, Walras and Pareto, to whom, I think, Wicksell should be added.”17 By that tradition, he wrote, he meant a study of the interrelationships of economic variables. His model of general equilibrium is complex. He assumed pure competition and an equilibrating process that has an intra-day or intra-week non-virtual phase. But he then made assumptions that ensure that equilibrium prices do not appreciably differ from what they would be if the model were purely virtual. He used the counting of equations and unknowns to establish existence, but tried to transcend that approach in his dynamics and capital theory. His summary statement explains why he did not present proofs of stability and uniqueness:
Since we shall not pay much attention to the process of equilibration which must precede the formation of the equilibrium price, our method seems to imply that we conceive of the economic system as being always in equilibrium.18
Other economists have also preferred not to deal with the complexities of non-virtual behavior, but have not wanted to make the highly unrealistic assumption that their model is always in equilibrium. They have chosen a virtual tâtonnement because it does not have the problems for stability caused by path dependency, but it also requires that account be taken of the question of existence in a virtual environment. On that matter, Karl Schlesinger prepared the way by showing that equality of numbers of equations and variables is neither necessary nor sufficient.19 He showed that the counting procedure is faulty because it does not take account of free inputs into economic activities, such as sunshine, air, and rain, and it lacks the necessary assumption that the variables are non-negative real numbers. Those results drew attention to the fact that a foundation had not been established for the construction of virtual competitive models. If existence could not be proven, it could not be asserted that the variables could possibly reach a general equilibrium. Stability and uniqueness require existence.
Abraham Wald, in his research on that question, confronted difficult problems, which he resolved in the usual way, namely by assuming they do not exist.20 First, he recognized that if utilities are interdependent, when the price of one commodity changes, the demands for related commodities could increase, decrease, or be unchanged, as is true in the real economy. The equations of general equilibrium, and therefore the question of the determination of whether equilibrium exists in a virtual or a non-virtual model—whether or not all markets can be in equilibrium simultaneously —are then extremely complicated. His procedure was to assume that the utility or satisfaction a consumer obtains from the use of any commodity is independent from that consumer’s use of any other commodity; for example, the satisfaction derived from the cereal in a bowl depends only on the cereal and is independent from the satisfaction derived from sugar and from milk. Second, he abstracted from economic growth. Third, he assumed pure competition. Fourth, he assumed a virtual tâtonnement. Fifth, he incorporated into his model the limitations on prices that Schlesinger had imposed, among others. Wald established uniqueness of equilibrium in the miraculous case in which all commodities for all consumers are gross substitutes. He also established the existence of equilibrium in the highly restricted special case of his model. Presumably, Pilkington would not accept Wald’s proof of existence as having sufficient realism and generality to be useful in economic inquiries.
After decades of effort to establish existence rigorously, the problem was studied and given a degree of resolution by Kenneth Arrow and Gérard Debreu.21 They assumed that their model is purely competitive, that it is virtual, that preferences are convex, and that production sets are convex and thus exclude economies of scale. The model has no behavioral, technological, and institutional mechanisms for the collection and dissemination of demand, supply, and pricing information, and therefore no equilibrating mechanisms. It has the timeless and static quality of the set theory in which is it expressed. The model is so unrealistic and limited in its generality that it has no practical applicability. Econometric general equilibrium models are not founded upon it, and cannot be, because it is purely competitive and virtual, whereas models used for policy and planning decisions deal with statistical data, most of which are generated in imperfectly competitive and non-virtual states of sectors of the real economy.
Pilkington asserts that “the existence of a general equilibrium is not in doubt.” The proof of existence to which he evidently makes reference is the one provided by the Arrow–Debreu model, which is the accepted neoclassical competitive model. Pilkington does not explain how he is able to reject neoclassical general equilibrium theory but apparently approve of the highly restrictive Arrow–Debreu model. Robert Clower has aptly described that type of model as science fiction, a judgment with which, despite his view of the existence issue, Pilkington agrees.22 He does not explain how he is able to reject neoclassical general equilibrium theory, but apparently approves of the highly restrictive Arrow–Debreu model.
The Current State of Affairs
Pilkington believes that no intuitively plausible stability mechanism has been suggested. Here is a summary of his reasoning. First, he is not in favor of the deus ex machina of a central auctioneer finding equilibrium prices. It is imaginary, unrealistic; “market economies are not run through a centralized auction.”23 Second, he does not believe the Walrasian reasoning that prices are changed by buyers and sellers and production by entrepreneurs in ways that lead the economy in the direction of a general equilibrium. Without further explanation, Pilkington rejects that intuitive notion of tâtonnement, describing it as providing “very little guidance,” a “barely coherent process of groping.” Third, he demonstrates the unlikelihood that general equilibrium could be reached by random chance, although he does not make clear why it is worthwhile to consider such a process, given its improbability and the theoretical results of the 1970s mentioned below. Walras and his followers, persons of genius who modeled non-virtual tâtonnement, would have rejected the suggestion that it can be evaluated by comparing its properties with those of a virtual random process. Fourth, Pilkington argues, ambivalently, that it does not matter whether a central auctioneer or the market system is modeled as the means of tâtonnement: “what difference does it really make?” he asks. Let the reader decide for himself or herself, he recommends, based on his or her experiences with auctions, the stock market, and shopping.24 Pilkington does not really advocate that policy, which would lead nowhere.
In actuality, Pilkington’s foregoing reasonings are criticisms of stability theorizing, so his challenge is presented to theoreticians.
This essay is meant only as a challenge to general equilibrium theory. To those championing the theory, a request: make a case for the departure from randomness that is intuitively plausible. … Until then, GET should be seen for what it is.
If stability is not in general a feature of neoclassical models, what, Pilkington asks rhetorically, is their usefulness?
In fact, theorists see non-virtual and virtual general equilibrium theory for what it is. The research of Hugo Sonnenschein, Rolf Mantel, and Debreu shows, in Pilkington’s words, that “the general equilibrium framework could not be used to infer a unique equilibrium.”25 That research also shows that stability cannot be proved to be a feature of neoclassical models, with which Pilkington agrees, but stability is essential for a model to be a useful functioning system. Thus, Pilkington’s essay repeats the answer that Sonnenschein, Mantel, and Debreu have given: microeconomic general equilibrium theory “reached a dead end: no general results beyond existence of equilibrium were possible.”26 As for the quality of the existence proofs, this letter has described it unfavorably.
Pilkington’s Challenge
If believers or non-believers want to accept Pilkington’s challenge to provide what he would accept as an intuitive modeling of stability, they will have to improve upon the theorizing of the great economists of the last 145 years. Today’s theorists would find that very difficult. It would require a degree of realism and comprehensiveness of scope and detail that go far beyond that of current econometric models. It would require modeling markets with non-virtual and virtual tâtonnements, purely competitive and imperfectly competitive markets, existing financial institutions and capital markets, and a variety of other existing determinants of economic growth in a single system. It would require modeling existing ways of information collection and dissemination, the large and diverse presence of government, and import and export sectors. It would require a moving equilibrium, interrupted by parametric changes and economic fluctuations.
Existing general equilibrium modeling does not do those things, and Pilkington therefore asks, “Does this mean we just have to scrap economic theory?” He replies, “No,” but his real answer is yes—scrap the part that is general equilibrium theory; that would be very helpful. “[E]conomics would be a far more successful discipline if it studied aggregate behaviors that are easily observable in the wide range of economic statistics that we possess,” and would enable more sophisticated policy discussions.
Fortunately, economists do precisely what Pilkington recommends, and as a result are able to make many sound prescriptions and predictions. Achieving more relevancy for general equilibrium theory is desirable, of course, but that would require more of the kind of work described above on modeling the features and interrelationships of economic variables. Given the magnitude and nature of the task, even with drastic simplifying assumptions, Pilkington’s plea for an acceptable intuitive model should not be expected to be fulfilled anytime soon. That means that his prescription is, for the present, the right one.
Donald Walker
Philip Pilkington replies:
I am honored that Professor Walker has responded to my essay. He is among those whom I counted as general equilibrium specialists and economists with strongly analytical gifts. These rare but subtle birds, as I pointed out, are far more attuned to the deficiencies with general equilibrium theory (GET) than their workaday economist colleagues. Yet, as can be confirmed from my opening paragraphs, it is to workaday economists—to the New Classicals, the New Keynesians, and everyone in between—that the critique is mainly addressed.
This accounts for many of the supposed inconsistencies that Walker has found in the essay. I state in the main body of the text that I do not doubt the existence of a stable equilibrium in an abstract GET model. Yet in footnote 4, I reference the Sonnenschein–Mantel–Debreu theorem and even point out that it was originally discussed by G. L. S. Shackle. I state that no less than Frank Hahn thought that this overturned microfounded economic models. What accounts for this discrepancy? The body of the text is written for the general economic audience, and the footnote is written for Walker and his colleagues. To dust off some old terms from Thorstein Veblen, the body of the text was designed to be exoteric and the footnote esoteric.
I am, for my sins, a workaday economist. I am interested in applied models and that is how I approach GET. The questions GET raises for me are not really ones of stability, of whether a given solution is virtual or non-virtual, or anything of this sort. Rather my core question is as follows: does GET provide a framework on which I can build a coherent idea of the really existing market economy that I inhabit so as to analyze it, try to understand its deep working, and even—dare I say it—venture some falsifiable predictions?
This is no minor question. As I state in the opening paragraphs, all models that utilize general market clearing as a mechanism for explanation rely implicitly on GET. This includes increasingly controversial modern microfounded dynamic stochastic general equilibrium models. But it also includes neo-Keynesian models such as the IS–LM and the Solow growth model and its offshoots.27 I would go so far as to say that the vast majority of analytical techniques that economists—micro or macro—are taught today rely on GET as they rely on the assumption of general market clearing. Unless someone can explain why I am wrong, I see the latter as being simply an assumption of general equilibrium.
This is where Walker’s approach and my approach diverge sharply and where I think that he misunderstands not only my intentions but also my methodology. Walker continues to occupy what the philosopher of science Hans Albert called the model Platonism of mainstream economic theory.28 Walker believes that if only the general equilibrium theorists can produce an abstract model that converges on equilibrium, we can all go home happy. I am not interested in this. I am interested in the really existing economy we inhabit—the one that I produced concrete, empirical estimations for. I want to explain these empirical estimations in a way that makes good logical sense.
In my essay, I have shown the number of transactions that take place in the United States economy in any given month. Day by day, economists in academia, in government departments, and in central banks assume that these transactions are settled through general market clearing. As already stated, general market clearing is effectively synonymous with GET. This is why I have provided an estimation of the probability of converging on this general market clearing by randomness. Randomness here is our base case. My cri de cœur to these hundreds, perhaps even thousands of economists is simple: explain your assumption! You are assuming, as a matter of fact, general market clearing processes, but you have not explained how you have gotten there!
This is an enormous analytical sin. We are told our starting point—disequilibrium. We are told the terrain we must cover—billions of transactions per month. And we are told our end point—general market clearing. But we are never told how we get from starting point to ending point.
“How did Bill get from Dublin to London in only two hours when all the airports were closed, and no planes were allowed to fly?” I ask you. To this you respond: “Well, Bill was in Dublin at 1pm and he appeared in London at 3pm.” “I am aware of that,” I respond, frustrated, “but how did he get from Dublin to London?” “Oh,” you are taken aback momentarily by my insolence, “why he traveled there, through space of course.”
We are being bamboozled through perpetual deferral. The economists who assume market clearing are making an extraordinary claim—this is what my analysis of randomness shows. But they cannot give a coherent explanation of how this extraordinary claim is realized. Again, I do not mean this from a highly abstract Platonic modeling perspective. I mean this concretely—I want to know how Bill got from Dublin to London in two hours with no access to flights.
This is where, although our assessment of GET is somewhat similar, Walker and I diverge. He believes that “economists do precisely what Pilkington recommends, and as a result are able to make sound prescriptions and predictions.” My contention is that they do not. Almost all their analytical techniques assume market clearing. And what my probabilities suggest is that, given the enormity of the reality that they are trying to wrap their arms around, they need to justify this assumption in concrete terms and with respect to the empirical data. Without that, they should stop using models that make use of general market clearing. But that would require rejecting much mainstream economics. Until then, they are babbling alone and in the dark.29
