How Mathematical Economists Overreach

by Mario Rizzo

In recent months there has been a discussion both in the traditional media and in the blogosphere about why orthodox macroeconomics failed to predict or explain the financial crisis and the subsequent Great Recession. Some of that discussion focused around Paul Krugman’s criticism that economics mistook  (mathematical) beauty for truth. Subsequently, there was a further discussion about the role of mathematics in economics.

Of course, this is a big topic. My task here is only to investigate, by means of a simple example, three claims made for the superiority of mathematics over ordinary (natural) language.

This example comes from a very interesting article, “Austrian Marginalism and Mathematical Economics” by Karl Menger. Karl Menger was the mathematician son of Carl Menger, one of the three pioneers of the marginal revolution and the founder of the Austrian school of economics.  (The article is a chapter in J. R. Hicks and W. Weber, Carl Menger and the Austrian School of Economics.)

Karl Menger evaluates some claims by mathematical economists in the context of the Principle of Diminishing Marginal Utility. He states the idea in words:

“For each good, the utility of a larger quantity is greater (or at any rate not less) than that of a smaller quantity, whereas the marginal utility of the larger quantity is less (or at any rate not greater) than that of the smaller.”

(For our purposes here let us disregard the question of the cardinal measurement of utility.)

Compare this to the standard formulations in terms of a twice-differentiable utility function.

U=f (q) where f’(q)> or = 0  and f”(q)< or = 0.

Some economists claim that the mathematical formulation is: (1) more general, (2) more explicit and (3) more precise.

Is it more general?

No. Actually, the verbal formulation is “more general since it is valid even if there are places where the function does not admit a second derivative and its graph has no curvature, whereas at such places the mathematical formulation fails to assert anything.”

Does the mathematical formulation make the assumptions underlying the Principle more explicit?

There us a lot of ambiguity in this question. Do the proponents of this view mean the assumptions underlying the verbal formulation or do we mean the assumptions underlying the mathematical formulation? Clearly the meaning must be the mathematical formulation.

Here again the answer is no.  The assumptions of continuity and differentiability of the utility function are made “as though these properties were matters of course, whereas they are nothing but prerequisites for the application of classical [mathematical] analysis…”  Far from bringing the tacit assumptions of the verbal formulation to the fore, the mathematical adds new assumptions in an almost casual and therefore implicit manner.

Is the mathematical formulation more precise?

No.  Precision must be defined relative to what the analyst desires to express. Each of these formulations is equally precise. The only difference between the two formulations is that the mathematical is restricted to cases in which the “functions … are differentiable, and therefore have tangents (which from an economic point of view are not more plausible than curvature)…”

Having said all of this, there is an additional consideration that goes beyond those considered by Karl Menger. This is the very strange “essentialist” idea that the true Principle of Diminishing Marginal Utility is the mathematical one.  Consider this nuance: If an economist says the virtue of the mathematical approach is that it makes the assumptions underlying the Principle more explicit, he is begging the question. This is because he is not referring to anything that exists independently of his own formulation.

Could he really be saying that the mathematical formulation of the Principle is more useful?  This is probably what the mathematical economist means. But when words like “useful” are being bandied about, we must have a pragmatic meaning in mind. Useful for what? For making more elaborate mathematical models? Obviously. But why do we want them? What are they useful for?

At this point we come to some fundamental issues. I cannot treat them all here. However, there is the often-ignored three hundred pound gorilla in the room.

We have data; we have implications of a model. The data have been processed and massaged to fit the general needs of the theoretical approaches. The implications of the model are mostly abstract; they must be interpreted as corresponding to phenomena that are observed in everyday life.

The most important issue is what counts as an explanation? More exactly, what shall we accept as an explanation? (This is the gorilla.) What we shall accept is a decision.

Some of us are implicitly asking for an explanation that avoids or minimizes or de-emphasizes assumptions that are purely mathematical. As Karl Menger says, “…they are nothing but prerequisites for the application of classical analysis and not based on facts.”  In other words, they are for the convenience of applying the method.

There is a certain self-contained quality to al this. (I shall not say circularity.) Mathematical theory transforms the phenomena of everyday life (Schutz and Luckmann’s “life-world’) into another form and then explains those data.

I hope the reader can see that in itself there is nothing “wrong” with this. However, it is a different endeavor. It is not doing the same thing as economists of old did in ordinary language  but doing it better or more generally or more explicitly or more precisely.

It is doing something else. It is changing the subject to fit a method.

Is there not a role for answering different questions with the precision (and so forth) most appropriate to them?

Yes, this is just what Aristotle famously says in his Nicomachaen Ethics:

“Our discussion will be adequate if it has as much clearness as the subject-matter admits of, for precision is not to be sought for alike in all discussions, any more than in all the products of the crafts.” (Book I, Chap. 3.)

25 thoughts on “How Mathematical Economists Overreach

  1. Is there not a role for answering different questions with the precision (and so forth) most appropriate to them?

    Who would answer no?

    As far as I’m concerned, the point is that both the mathematical and verbal formulations are imprecise because both take some concept of utility as given.

    What is utility?

  2. Two problems:

    1) we cannot ever know if we have enough data or the right data or enough precision in the data when dealing with a complex system. Any of these will result in false predictions.

    2) we forget that, as the information theorist Pierce once said, “Mathematics is a precise approximation of reality.” We forget the “approximation” part.

    So with math being inherently an approximation, and with the kind of math necessary for a complex system like an economy, we see that prediction is literally impossible (in the Newtonian physics kind of way people want from their predictions).

    In economics, math is only useful to create models. It should not be mistaken for anything even remotely real.

  3. Troy,

    Well put. But what’s the difference between your comment and the comment below?

    * * * *

    Two problems:

    1) we cannot ever know if we have enough experience or the right experience or enough precision in the experience when dealing with a complex system. Any of these will result in false predictions.

    2) we forget that, as the linguist Ecreip once said, “Language is a precise approximation of reality.” We forget the “approximation” part.

    So with language being inherently an approximation, and with the kind of language necessary for a complex system like an economy, we see that prediction is literally impossible (in the Chomskian kind of way people want from their predictions).

    In economics, language is only useful to create descriptions. It should not be mistaken for anything even remotely real.

  4. I doubt you are referring to modern mathematical economics. Here, the utility function is an ordinal, a monotonic transformation of the underlying preference preordering (a similiar argument applies to production). Diminshing marginal utilities is a non-ordinal property. It is rooted in weak convexity assumptions we impose on preferences. Further, if you choose nonconvex consumption sets (not to confuse with the convexity of preferences) by assuming indivisibilities or non-additivity you cannot derive continuous utility functions. Nonconvexities in general allow for all kinds of discontinuities. The good think about the axiomatic method is that you know where these irregularities come from. Further, the convexity assumption is important in the existence proof. Either preference preorderings of agents are convex or we need a continuum of agents or large but finite number of agents to proof existence. Since existence is not generally proved by counting variables to determine and equations (equilibrium conditions) we need Debreu’s toolset. We cannot get to the nature of these relationships without math. HAYEK used the neoclassical apparatus (indifference curves, production possibility frontiers, and the like) to counter Böhm-Bawerkian theories of intertemporal choice (Utility analysis and Interest, I think 1936). I agree, though, that economic interpretation should dominate mathematical inquiry in economics.

  5. AMV,

    I used a simple example — one that still dominates intermediate microeconomics texts and is taught to many more students than any other form of “mathematical economics.” However, every form of mathematics has its own particular assumptions and restrictions that make it different from the analogous verbal formulations. Difference, not better or worse, is my central point. My argument is against methodological exclusivism.

  6. Great post.

    I’d like to add the two Rothbardian main arguments against mathematical economics:

    1 – It violates Occam’s Razor

    2 – The concept of function implies causal regularities. But we don’t have them in economics. Ignoring the discussion if it makes any sense to express utility in cardinal statements, U = f (q) is always changing with endogenous learning or exogenous shocks. A pretty deleterious method for a science that neeeds to study a market process in real time.

    AMV remembered Hayek’s Pure Theory. The biggest shortcoming in it, IMHO, are those unintelligible “neoclassical” diagrams he used. If he had used some clear verbal logic, the book would be a lot better.

  7. Economists are forever equating apples to oranges. Wages cannot be equated to the marginal productivity of labor because one is an apple and the other, an orange. This is a most fundamental point in the early Austrian tradition, and that is why real Austrians refuse to use mathematics.

  8. Perhaps the problem is concerned more with a sociology of economics. That is, a study of the bias among the mainstream economics profession in terms of its resistance not only to criticism of the mathematical method, but suggestions for any alternative methods of analysis (primarily) in place of theirs.

    In the late 19th century it could be claimed, perhaps, by the advocates of the mathematical method in economics that the earlier critics just did not have a reasonable enough knowledge and background in the use of the tool to fully appreciate it.

    This was the charge that people such as Pareto raised against the “literary” economists of his time.

    But in the second half of the 20th century criticisms were made concerning the limits, uses and abuses of mathematical economics by those who could not be considered “ignorant” or “uninformed” about the method or its applications. I would mention (randomly): George Stigler, Kenneth Boulding, Fritz Machlup, Oskar Morgenstern, George Shackle, to just name a few who would be considered respected members of the profession.

    Yet, virtually all such criticisms have seemingly been just so much “water off a ducks back.”

    I am currently reading Thomas Sowell’s new book, “Intellectuals and Society.” It is a biting attack on the mind set and biases of too many politically motivated intellectuals in society.

    Sowell points out (as he has in a number of other writings) that in many fields theoretical models and empirical assumptions are confronted with a “reality” from which the thinker cannot escape. (Did the constructed bridge stand or collapse? Did the product manufactured actually find a buying public that were willing to pay a price for it that covered the costs of it’s production.)

    But in the world that academic and many public opinion intellectuals spend out their lives there is no similar “reality check.” There is only the praise or rejection by those who think like oneself, and the often reinforcing of “prejudices” about problems, solutions and ways to get from one to the other.

    What is the reality check for most economists? Being accepted for academic publication by editors and reviewers who judge the value and methods used in a submission by the standards that those editors and reviewers already — “a priori” — consider to be “scientific” and therefore correct.

    And dissidents don’t seem to last long in this setting. Maybe some will recall that Robert Clower was for a brief time the editor of the “American Economic Review.” For that short period the articles in the AER were: (a) often readable (in terms of both style and content); (b) were sometimes outside of the Neo-Classical model mentality; and (c) actually, were often interesting to read.

    His editorship did not last very long. He just did not fit what Sowell has often referred to as the “vision of the anointed.” That high priesthood of the math-econ tribe about which Axel Leijonhufvud satirized many years ago.

    Richard Ebeling

  9. Richard,

    I believe that much of what you and Sowell say is correct. But just like public choice explanations are a supplement to critiques of the general-welfare effects of policies, I think the “sociological” explanation of the mathematization of economics must be a supplement to the more direct analysis of its nature and consequences.

  10. Mario:

    I agree with you. The logic of a theory or a method should be judged and debated on its own merit. And I, too, have often drawn people’s attention to Karl Menger, Jr.’s analysis in the article from which you quoted.

    And as much as I find the Public Choice arguments logically and factually compelling, they often fall short in one important way: they too frequently leave unchallenged the theoretical arguments of those whose self-interested motives they analyze.

    But, it is also the case that much of the resistance to non- or less-mathematical approaches in economics can be traced to the types of biases and prejudices that Sowell, for example, has emphasized.

    His analysis of how many intellectuals ridicule opposing ideas rather than actually confront them certainly have their counter parts in the mainstream rhetoric about, say, Austrian Economics.

    How often have many of us heard it said that Austrian Economics is a “faith” or “religion”? That it is anti-scientific, or merely an ideological rationale for “right-wing” policies. That is is just “fluff” or “talk” (meaning that a “real” argument is economics can only be one clothed in the mathematical garb)?

    And these statements are presumed to be sufficient to drive the Austrian from the economics temple of respectable debate and discourse.

    In 1998, Paul Krugman said, “A few weeks ago, a journalist devoted a substantial part of a profile of yours truly to my failure to pay due attention to the ‘Austrian theory’ of the business cycle—a theory that I regard as being about as worthy of serious study as the phlogiston theory of fire.”

    Interestingly the same exact charge was leveled against Ludwig von Mises BEFORE the First World War. Mises recounted in that at a meeting of the Verein fur Sozialpolitik before 1914, during conversations he had with members of the German Historical School:

    “One of these gentlemen remarked that a colleague of his had asked whether I was not also an adherent of the phlogiston theory [in reference to Mises’ monetary theory]. Another gentleman suggested that he considered my “Austrianness” to be a mitigating circumstance; with a citizen of Germany he wouldn’t even discuss such questions . . .”

    The more things change . . .

    Richard Ebeling

  11. “What is utility?”

    Satiation of wants. Wow, rocket science.

    “In economics, language is only useful to create descriptions. It should not be mistaken for anything even remotely real.”

    Good luck trying to communicate *anything* without language. Maths is one too. No one ever said language is what makes the descriptions true, the argument is over which language best conveys them… understood now?

  12. Pete Boettke is perplexing over the same basic issue at the Coordination Problem. I find the exchange here between Mario and Richard very useful.

    There was a long-running debate between Kirzner and Lachmann over whether Austrian Economics should have been designated a field of study. Kirzner said “no,” because he wanted to be integrated with the profession. Lachmann said “yes,” because he wanted to develop a research program.

    In theory, Kirzner was right; in practice, Lachmann was correct. My model is Public Choice.

  13. The philosophical background most relevant to this debate would be discussions about the relationship between formal logic and other modes of reasoning.

    For example, when Mario writes: ” The assumptions of continuity and differentiability of the utility function are made “as though these properties were matters of course, whereas they are nothing but prerequisites for the application of classical [mathematical] analysis…” Far from bringing the tacit assumptions of the verbal formulation to the fore, the mathematical adds new assumptions in an almost casual and therefore implicit manner.” the philosophical question of interest is: are these formal assumptions essentially conservative, that is allow no inferences to be made in the mathematical model which are not valid in the language model?

    Let me give a brief explanation here, using a simple example from formal logic.

    There is a well known construction or translation of the english “if, then” into material implication which illustrates a great deal about formalization and conservative inference patterns.

    (A materially implies B is equivalent to (not) A or B)

    The formalization of “if, then” as the material implication follows from 3 observations and 2 conditions.

    The conditions are:

    1. The formalization has to be truth functional, the truth value of “if A, then B” can only depend on the truth value of A and B.

    2. The formalization has to preserve valid inferences: if A, then B is valid, the A materially implies B has to be valid.

    The three observations are:

    1. if A, then A is true for all truth valuations of A.
    2. if A, then B is false for A is true and B is false.
    3. if A, then B does not logically imply if B, then A.

    The paradoxes of material implication as a translation of if the are well known.

    What economists do reasonably well in modeling is find an abstraction of a social exchange which is truth-functional.

    What they don’t generally do very well is to test whether their truth-function translation preserves inferences, let alone ask whether the model is essentially conservative.

    Here is simple example: the prisoner’s dilemma is not an abstraction from a real prisoner’s problem which preserves the inference that prisoner’s rat out each other. The latter is simply not true. In this case, we have a simple truth functional translation of strategic talk which doesn’t preserve valid inferences in home language.

  14. This is a great discussion. The cited passages from K. Menger are important. michael webster makes a great point which I do not recall seeing before. The paradox of material implication provides a really good and vivid illustration of his point. Thanks for that, michael.

    Oskar Morgenstern has a great article on mathematical economics in which he says that you cannot really make a general argument against math since someone might makeup some new math (e.g. game theory) that gets around your objection. That argument missed the importance of Hayek’s diagonal argument from The Sensory Order: You cannot eliminate words referring to mental categories from your description of human action without significant loss of information. Hayek´s point supports Mario’s remark that zeal for math changes the game, “changing the subject to fit a method.” Right on.

    Still, I think Morgenstern was right to caution us against general arguments to the effect that “math is bad.” (Mario wisely avoided such claims.) In particular, I think “literary” economists can get some milage from computable economics. That stuff takes seriously all the post-Goedel stuff on how certain things in math are not “computable.” In mathematical economics we often have agents instantaneously computing things that are either “hard” or impossible to compute. If we ramped up our mathematical sophistication to the level reached in computable economics, then we would have the mathematical tools required to show that a lot of mathematical economics is magical thinking and not in the least rigorous.

  15. Is there a suggested book about computational economics? It seems a nice topic but I know nothing about it. I plan to grasp the basics, and I have some computer science / math background.

    A friend of mine has a journal paper (JASSS) on ABM, but she is more focused on statistical prediction than on market process, so although I got curious, I prefer a different perspective.

    I’ve appreciated Axtell’s “The complexity of exchange”, and I had the impression to know some of those things beforehand, because many things had a straightforward (maybe my delusion) Austrian interpretation.

    Well, despite my Austrocentric view of the world, there seems to be a lot there to catch my attention.

  16. Pietro,

    For my money, the best source is

    Velupillai, Vela. ed. 2005. Computability, Complexity and Constructivity in Economic
    Analysis. Oxford: Blackwell.

    See also Velupillai, Vela. 2007. “The Impossibility of an Effective Theory of Policy in a Complex Economy,” in Salzano, Massimo and David Colander (eds.),Complexity Hints for Economic Policy. Milan, Berlin, and elsewhere: Springer.

    The latter links to policy issues and is a natural for Austrians.

    I have tried to convey some sense of the stuff in several articles.

    “Computable Entrepreneurship,” Entrepreneurship Theory and Practice, 2008, 32(5): 919-926. There is an ungated version somewhere on the web. Sorry I don´t have that at the moment.

    “Thinking Impossible Things: A Review Essay on Computability, Complexity and Constructivity in Economic Analysis,” Journal of Economic Behavior and Organizaiton, 2008, 66: 837-847.

    “All That I Have to Say Has Already Crossed Your Mind,” with Barkley Rosser, Metroeconomica, 2002, 53(4): 339-360.

    I briefly connect Velupillai 2007 to macro on pp. 29 & 30 of my BRACE draft:

    Click to access BRACE.pdf

  17. Thanks. I think some of these papers are accessible from my PC. I don’t know why, but my engineering department subsidizes JSTOR, JoET, JoME, JoPE and I can free ride on it. 🙂

  18. Very interesting read.

    I have a couple of more practical problems with the use of math in economics in mind, mostly with teaching economics. This is in no way a complete list of my complaints about math but these are the things that bother me the most.

    1. a) One comment I hear often from students about economics in general is that “it’s too theoretical”. When you dig you figure out they often mean “it’s too mathematical.” It adds a level of abstraction to economics that we don’t really need..

    b) My experience is that “it’s too theoretical” sometimes mean they’re skeptic about economics because of it’s use of mathematics. The use of math to prove anything and it’s exact opposite make economics look like a purely rhetorical discipline.

    2. a) Let’s be honest, the instruments are hard to master and the professor winds up wasting too much time explaining the tools, and the students focus too much on the tools and their economic interpretation.

    b) No matter how little the professor introduces math in his class, the teaching assistants will focus exclusively on that in his sessions. Then we end up with classes full of students that are pretty good in anything mathematical, but are unable to explain the economic significance of what they just did.

    3. Regarding research I have one serious annoyance; the second generation pretty much always forgets the reservations the people who developed the tools expressed (sometimes only implicitly) about using their math as descriptive tools. This is so widespread and systematical that there is probably something inherently incompatible with the the use of math in economics and scientific incentives…

    Just my two cents.

  19. […] There has been a lot of negative talk about the maths in economics – like a huge amount.  Just look at these links, some of them are poor and reactionary, but some of them are excellent and the last two are my favourite (*, *, *,*,*,*). […]

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