Arrow-Debreu obsession
from Lars Syll
I’ve never yet been able to understand why the economics profession was/is so impressed by the Arrow-Debreu results. They establish that in an extremely abstract model of an economy, there exists a unique equilibrium with certain properties. The assumptions required to obtain the result make this economy utterly unlike anything in the real world. In effect, it tells us nothing at all.
So why pay any attention to it? The attention, I suspect, must come from some prior fascination with the idea of competitive equilibrium, and a desire to see the world through that lens, a desire that is more powerful than the desire to understand the real world itself. This fascination really does hold a kind of deranging power over economic theorists, so powerful that they lose the ability to think in even minimally logical terms; they fail to distinguish necessary from sufficient conditions, and manage to overlook the issue of the stability of equilibria.
Almost a century and a half after Léon Walras founded neoclassical general equilibrium theory, economists still have not been able to show that markets move economies to equilibria.
We do know that — under very restrictive assumptions — equilibria do exist, are unique and are Pareto-efficient. After reading Buchanan’s article one however has to ask oneself — what good does that do?
As long as we cannot show, except under exceedingly special assumptions, that there are convincing reasons to suppose there are forces which lead economies to equilibria — the value of general equilibrium theory is negligible. As long as we cannot really demonstrate that there are forces operating — under reasonable, relevant and at least mildly realistic conditions — at moving markets to equilibria, there cannot really be any sustainable reason for anyone to pay any interest or attention to this theory.
A stability that can only be proved by assuming “Santa Claus” conditions is of no avail. Most people do not believe in Santa Claus anymore. And for good reasons. Santa Claus is for kids, and general equilibrium economists ought to grow up.
Continuing to model a world full of agents behaving as economists — “often wrong, but never uncertain” — and still not being able to show that the system under reasonable assumptions converges to equilibrium (or simply assume the problem away) is a gross misallocation of intellectual resources and time.
And then, of course, there is Sonnenschein-Mantel-Debreu!
So what? Why should we care about Sonnenschein-Mantel-Debreu?
Because Sonnenschein-Mantel-Debreu ultimately explains why New Classical, Real Business Cycles, Dynamic Stochastic General Equilibrium (DSGE) and “New Keynesian” microfounded macromodels are such bad substitutes for real macroeconomic analysis!
These models try to describe and analyze complex and heterogeneous real economies with a single rational-expectations-robot-imitation-representative-agent. That is, with something that has absolutely nothing to do with reality. And — worse still — something that is not even amenable to the kind of general equilibrium analysis that they are thought to give a foundation for, since Hugo Sonnenschein (1972) , Rolf Mantel (1976) and Gerard Debreu (1974) unequivocally showed that there did not exist any condition by which assumptions on individuals would guarantee neither stability nor uniqueness of the equlibrium solution.
Opting for cloned representative agents that are all identical is of course not a real solution to the fallacy of composition that the Sonnenschein-Mantel-Debreu theorem points to. Representative agent models are — as I have argued at length here — rather an evasion whereby issues of distribution, coordination, heterogeneity — everything that really defines macroeconomics — are swept under the rug.
Instead of real maturity, we see that general equilibrium theory possesses only pseudo-maturity.
For the description of the economic system, mathematical economics has succeeded in constructing a formalized theoretical structure, thus giving an impression of maturity, but one of the main criteria of maturity, namely, verification, has hardly been satisfied. In comparison to the amount of work devoted to the construction of the abstract theory, the amount of effort which has been applied, up to now, in checking the assumptions and statements seems inconsequential.
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Real-World Economics Review Blog 
A challenge for actual living real-world economists: Under what conditions (i.e., what set of actually feasible legal and public pollicy limitations) will an actual existing economy tend toward a sustainable equilibrium, with “equilibrium” defined as “a situation in which every person’s needs for food, shelter, security, health, and meaningful work are provided for in the present and for the foreseeable future”? Please show your work. Extra credit if you describe a sustainable equilibrium that is attainable within 50 years on the actually existing planet Earth, starting now.
The answer is NEVER, for neither economies nor markets ever tend towards ‘equilibrium’. Once one throws out the mathemagics behind the construction of ‘demand’ curves, that improbability of ‘equilibria’ becomes self-evident. With it also goes all of what is currently called ‘Welfare Economics’ as well as all macro-economics based upon micro-economic theory.
The better question is whether economies can be put upon stable paths which would include ” situations in which every person’s needs for food, shelter, security, health, and meaningful work ARE MOSTLY BEING MET for in the present and for the foreseeable future”, though ‘meaningful work’ implies moving away from ‘market-oriented’ jobs and would governmental payments for those as providing what would have to be considered as public goods either available at low costs or no costs to users. That implies a social dimension to production even when the goods themselves are produced for-profit (always to government standards as buyers who distribute these goods to users).
Challenging the current economic orthodoxy instantiated in the extant political economy is like a medieval peasant challenging monarchical, ducal and lordly authority temporal and spiritual. Chances of success are zero without a revolt. In turn, it is rare that peasant or mass revolts succeed. Without a political revolution, there can be no people-directed change, as opposed to elite management and lack of real change in the political economy. Do you see the people around you rising up en masse to conduct a popular revolt? No? Well then, stop expecting anything to change by the mere non-act of theorizing in rarefied circles. That would be my advice, even to myself, as all I do is theorizie in rarefied or at least obscure circles.
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Insightful. I don’t believe history repeats itself in exactly the same way. So looking to past “peasant or mass revolts” may not be a very good guide for humanities current situation. You might find the following interesting books (one of my daughter’s professors wrote the first one) for giving some context to our times.
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Pessimism like optimism is a choice. I find it best to neither be a blind optimist nor a blind pessimist, but to face reality with eyes wide open but with the wisdom to see the arc of history pointing to positive side of our unrevealed future. There are many untold stories of positive progress.
Yes. Neoliberalism –the kind which underlies the every-man-for-himself capitalistic ethos– has been confused for what economics is about, as if exchange is the heart of economic activity, and markets all all.
Economics is always and everywhere about how we provide for ourselves not merely as individuals but altogether.
As a founding member of Cartographers Without Borders, I reserve the right to theorize in rarefied or even non-rarefied circles. Who knows, it might spark an interesting conversation, if not an actual revolt. “The world is flat!”, say leading economists. “Economics is revolting!”, say leading ordinary people. “What the hell are they even talking about?”, says everyone else. “My 401K is doing great!”, say the One Percent. “Let’s just keep going.” Second only to the fact that many people appear to think that complaining about conventional economics is a non-non-act (when it is by this point obviously a non-non-non-act), the. most annoying thing about many people is that they refuse even to argue about the purpose of an economy, preferring the flavor of ad hominem attacks on dead or near-dead thinkers. Come on, economists! Why won’t YOU start the revolution? The dead horse of mainstream economics will not jump to its feet and resume pulling the cart, regardless of whether you beat it with a stick, a whip, or your thorny intellect. And (by the way) I am not theorizing. I am simply telling you what I think the purpose of economics should be and how we should start thinking about it.
Ikonoclast, nevertheless it can be done and quite simply so. Most of the ‘foundations’ of micro-economics can be shown not to exist and are readily and easily replaced [not that getting to there has been ‘easy’], opening up the profession to entirely new modes of analysis. Academic economists will have little choice but to follow once they realize that they cannot even defend how they construct the theoretical ‘demand curves’ they are so prone to use and build upon.
Sadly, I am a very slow thinker and even slower writer, Ikonoclast. Sorry ’bout that. Since I have no idea about how I can submit an article here, I will submit one long comment in early February.
Here are my major themes:
1. All agents of economic activity are producers-who-consume, whether these are persons of ‘firms’.
2. Objective benefits –that is, measurable value-in-particular-uses– underlies all use. Consumption always being usage (not purchase, albeit purchase is always for particular uses with measurable benefits, including monetary profits from production), there cannot be separate theories pertaining to consumers or producers in economics, albeit the maintenance and survival needs of life forms are hardly identical to those of ‘firms’. When all agents are producers-who-consume, we must construct theory which recognizes this fundamental reality and then integrates it into macro-economic theorizing built upon this new foundational structure. (In other words, the new economics integrates micro and macro into one, basic economic framework for both.
3. The mathematics of maximization found in most textbooks are incorrect most of the time because of the confusion between value-in-exchange and values-in-use. In fact, this confusion underlies the mathemagics on which consumer theory has been constructed.
4. The distribution of incomes/wealth is fundamental to establishing any and all macro-economic paths, including homeostatic ones, albeit our capacities for invention in technologies and techniques will always introduce new horizons as well as undertainties.
Please pardon the typos. I hope they do not take away from the sense of what I have said.
Iconoclast: After getting some sleep, I realized I should not that I use Achille-Nicolas Isnard’s equation of exchange x/y = px/py to reconstruct the income and substitution effects of Hicks in his Vale and Capital (and Slutsky’s earlier work), only to show that all analysis that proceeds from this equation of exchange equivalencies reveals that these effects are only the result of not permitting budgets to be flexible in response to price changes.
{I hadn’t realized that Isnard had used that equational form for equivalent exchange in his Traité des richesses back in 1781. When used in Cartesian x- and y- coordinate space, it yields loci whose coordinates (x, y =(px/py)x) describe every point on it and, in particular, every fixed monetary budget necessary to purchase any coordinate combination having equivalent values for X and Y as goods. In a sense, I feel I have re-invented the wheel by re-inventing an equation and using an exchange equation he published back then without my having known he had formulated it way back then.}
Thank you, Lars Syll, for focusing Arrow-Debreu General Equilibrium theory. All criticisms against its very restrictive assumptions are right. It is true that Arrwo-Debreu model cannot be supplemented by stability and uniqueness. Pretension of DSGE people that their theory has a microfoundation is flawed.
I have no intention to defend the model, but Syll and commentators’ comments seem to be forgetting one thing, i.e. without a theory that can replace A-D theory, economic theory does not and cannot change revolutionary. Neoclasscial mainstream remains mainstream. As a dictum says, it needs a theory to beat a theory. But, there is good news now. We have now a theory that can compete and even overcome A-D theory. The followings are excerpts from the Preface of our book Microfoundations of Evolutionary Economics:
«Representative agent models are — as I have argued at length here — rather an evasion whereby issues of distribution, coordination, heterogeneity»
But if you mention the Arrow-Debreu-Lucas models, and the Sonnenschein-Mantel-Debreu theorem, you must also mention the “three fables” from J.B. Clark:
https://www.aeaweb.org/articles?id=10.1257/089533003321165010
«Samuelson (1962) called three key “parables”:
1) The real return on capital (the rate of interest) is determined by the technical properties of the diminishing marginal productivity of capital;
2) a greater quantity of capital leads to a lower marginal product of additional capital and thus to a lower rate of interest, and the same inverse, monotonic relation with the rate of interest also holds for the capital/output ratio and sustainable levels of consumption per head;
3) the distribution of income between laborers and capitalists is explained by the relative factor scarcities/supplies and marginal products.»
The Arrow-Debreu-Lucas model assumptions are specifically designed to support the three fables/parables, and in particular number 3, and that is all about distribution. And Economics is all about “internal consistency” with those three fables/parables.
Blissex: The Collected Fairy Tales of Micro-Economics –all of which always end with “And they all lived happily ever after all of the time”– are the essential backstories which, er, ‘support’ the Macro-Economic Fairy Tale told by the Arrow-Debreu-Lucas model. Sadly, all of the tales are much less entertaining and edifying than those unexpurgated tales told by the Brothers Grimm. And, yes, John Bates Clark was a prolific teller of micro fables like those you’ve mentioned here.
Blissex’s account of Arrow-Debreu model is too politically focused. There is no big difference between his contention and that of neoclassical economists who argue that income determination by marginal productivity of each factor is the natural desirable state.
Marginal theory of income distribution is wrong, but we should show it wrong in a deeper level of theory than its mere political use. Politically oriented people (I read many of them in this blog) have a strong tendency to seize everything whether a theory (and economics as a whole) is useful or not for their political propaganda. This is something similar to try to compete Trumpists by spreading fake news.
To help at least in part to answer your question, I offer this from Lancelot Hogben the English zoologist, geneticist, medical statistician, and linguist, known especially for his many contributions to the study of social biology. In Mathematics for the Million, Hogben made extensive efforts to help us understand the social-historical situatedness of mathematics. This is a quote from that book.
“In the time of Diderot the lives and happiness of individuals might still depend on holding correct beliefs about religion. Today, the lives and happiness of people depend more than most of us realize upon correct interpretation of public statistics kept by Government offices. Atomic power depends on calculations which may destroy us or may guarantee worldwide freedom from want. The costly conquest of outer space enlists immense resources of mathematical ingenuity. Without some understanding of mathematics, we lack the language in which to talk intelligently about the forces which now fashion the future of our species, if any. We live in a welter of figures: cookery recipes, railway timetables,
unemployment aggregates, fines, taxes, war debts, overtime schedules, speed limits, bowling averages, betting odds, billiard scores, calories, babies’ weights, clinical temperatures, rainfall, hours of sunshine, motoring records, power indices, gas-meter readings, bank rates, freight rates, death rates, discount,
interest, lotteries, wavelengths, and tire pressures. Every night, when he winds up his watch, the modern man adjusts a scientific instrument of a precision and delicacy unimaginable to the most cunning artificers of Alexandria in its prime. So much is commonplace. What escapes our notice is that in doing these things we have learnt to use devices which presented tremendous difficulties to the most brilliant mathematicians of antiquity. Ratios, limits, acceleration, are no longer remote abstractions, dimly apprehended by the solitary genius. They are photographed upon every page of our existence.
In the course of the adventure upon which we are going to embark we shall constantly find that we have no difficulty in answering questions which tortured the minds of very clever mathematicians in ancient times. This is not because you and I are very clever people. It is because we inherit a social culture which has suffered the impact of material forces foreign to the intellectual life of the ancient world. The most brilliant intellect is a prisoner within its own social inheritance. An illustration will help to make this quite definite at the outset.”
No space to go into the illustration here. Suffice it is one we know it well under the title ‘the tortoise and the hare.’ But which Zeno and the others of his time who invented it named ‘Achilles and the tortoise.’
“You must not imagine that Zeno and all the wise men who argued the point failed to recognize that Achilles really did get past the tortoise. What troubled them was, where is the catch? You may have been asking the same question. The important point is that you did not ask it for the same reason which prompted them. What is worrying you is why they thought up funny little riddles of that sort. Indeed, what you are really concerned with is an historical problem. I am going to show you in a minute that the problem is not one which presents any mathematical difficulty to you. You know how to translate it into size language,
because you inherit a social culture which is separated from theirs by the collapse of two great civilizations and by two great social revolutions. The difficulty of the ancients was not an historical difficulty. It was a mathematical difficulty. They had not evolved a size language into which this problem could be freely translated.
The Greeks were not accustomed to speed limits and passenger luggage allowances. They found any problem involving division very much more difficult than a problem involving multiplication. They had no way of doing division to any order of accuracy, because they relied for calculation on the mechanical aid of the counting frame or abacus… They could not do sums on paper. For all these and other reasons which we shall meet again and again, the Greek mathematician was unable to see something that we see without taking the trouble to worry about whether we see it or not. If we go on piling up bigger and bigger quantities, the pile goes on growing more rapidly without any end as long as we go on adding more. If we can go on adding larger and larger quantities indefinitely without coming to a stop, it seemed to Zeno’s contemporaries that we ought to be able to go on adding smaller and still smaller quantities indefinitely without reaching a limit. They thought that in one case the pile goes on forever, growing more rapidly, and in the other it goes on forever, growing more slowly. There was nothing in their number language to suggest that when the engine slows beyond a certain point, it chokes off.”
We today see this without any effort. We are not smarter than the Greeks. We just live within a cultural heritage that provides us a mathematics of size and counting that the Greeks did not have. The economics profession was/is impressed by the Arrow-Debreu rule because the culture within which the profession and its practitioners reside literally forces attention on such artifacts as the Arrow-Debreu. Their understanding seems important and fits the world in which we live.