Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Monday, June 1, 2015

In the after-mathiness of discontent

If you're like me, you're probably still trying to wrap your mind around the debate on "mathiness" in the economics profession. I haven't put an inordinate amount of time into the project, but I have made an honest effort trying to understand the nature of Paul Romer's lamentations.

Just what is mathiness, anyway? I'm not sure that a precise and commonly-accepted definition exists. It's not surprising that such a catchy word was soon interpreted in myriad of conflicting ways. This is what happens with words.

To his credit, Romer soon recognized that the word he invented was being interpreted in ways he did not mean. He offers a sort of mea culpa here: Mathiness and Academic Identity. It is definitely a clarification (and I think, a softening) of his initial position. Here are what I take to be its two main points:
So my objection to mathiness is not a critique of the assumptions or structure of the models that others propose. It is a critique of a style that lets economists draw invalid inferences from the assumptions and structure of a model; a style that authors can use to persuade the reader (and themselves) to adopt conclusions that do not follow by the rules of logic; a style that tolerates wishful thinking instead of precise, clearly articulated reasoning. The mathiness that I point to in the Lucas (2009) paper...involves hand-waving and verbal evasion that is the exact opposite of the precision in reasoning and communication exemplified by Debreu/Bourbaki, and I’m for precision and clarity.
I wrote that the economists I criticize for using mathiness are engaged in a campaign of ACADEMIC politics, not one of national politics. Whatever was true in the past, the now fight is over ACADEMIC group identity. For example, one of the things that the people I criticize are campaigning for is a methodological restriction to models with price-taking. For them, price-taking is dogma. To make the case for this restriction, they are not presenting scientific arguments grounded in logic and evidence.
Hmm. He's critical of a style that lets economists draw incorrect inferences from the assumptions of a model. This is in contrast, I suppose, to all the other styles that permit people to draw incorrect inferences from the assumptions of their pet theories. He gives the example of Lucas (2009), to which I will return in a moment. The other charge is that some economists (them, not us of course) are "dogmatic." They campaign for and adopt methodological restrictions, like price-taking behavior, even when the logic and evidence does not permit it.

The two points above are linked to the same phenomenon: the apparent unwillingness of Romer's competitors in the field of growth theory to see the error of their ways and the superiority of his preferred approach. Mathiness is impeding scientific progress in the field of growth theory (and possibly throughout the economics profession as well).

What is the basis for these charges? As an outsider in the field of growth theory, it's hard for me to say (though I do have a paper in the area, which I'll talk about below). I think that Deitz Vollrath has a reasonable take on the issue here: More on Mathiness. Thankfully, we have Brad DeLong (Putting Economic Models in Their Place) doing his best to embellish the critique and apply it more broadly to macroeconomic theory--a subject more familiar to me.

Let's start with DeLong, who states:
He (Romer) seems, to me at least, to be very worried principally about two aspects of modern economic discourse. The first is to take what is true about one restricted class of theories and generalize it, claiming it is true of all theories and of the world as well.
Is there really any economist who behaves in this manner? Do we all not already know that our assumptions are provisional? That we may make one set of assumptions to address some questions and another set of assumptions to address other questions--that we have no grand unifying theory? So who, pray tell do these folks have in mind? Well, Paul Romer's thesis adviser, for one. Here, they quote Lucas (2009) who writes:
Some knowledge can be ‘embodied’ in books, blueprints, machines, and other kinds of physical capital, and we know how to introduce capital into a growth model, but we also know that doing so does not by itself [my italics] provide an engine of sustained growth. 
Now which parts of the quote above constitute a violation of scientific inquiry in your mind? Probably none. But I think it was the last part that got Romer's goat because he might have interpreted it to mean that "we know that Romer's models do not embody a mechanism that can provide an engine of sustained growth." I don't know--that's not the way I interpret the passage. Let's suppose Lucas had instead written "...but we also know that doing so does not necessarily provide an engine of sustained growth." Would that have been better? Can we all just be friends now?

I think it may be instructive to read the section from which the quote was lifted:

Is this really something to get so riled up over? Romer pg. 91 charges Lucas with making "untethered verbal claims"--an opinion he is of course free to hold. But I contend that it is just an opinion. Moreover, it's an opinion over the use of English, not math. It would have been far more useful and compelling, I think, if Romer had instead critiqued the model's internal logic and its ability to interpret the data. Isn't this the way science is supposed to progress? (Romer might reply that he's tried, but that his supervisor just won't listen because he's so dogmatic. I'll return to this possibility below.)

In any case, nowhere do I see evidence of DeLong's outlandish claim that Lucas is peddling an hypothesis he asserts to be true of all theories and of the world as well. But DeLong continues as follows:
Thus what Lucas claims must be true about the world as a matter of correct theory--that the big secret to successful economic growth cannot lie in creating and acquiring the kind of knowledge that gets "'embodied' in books, blueprints, machines..."--rests on the barely-examined decision to restrict attention to only a few kinds of models.
My goodness...this Lucas character...who does he think he is? You know, I have an idea. Why don't we actually go read the offending article and see what he's really up to? To whet your appetite, here's the introductory paragraph:

You know, call me crazy, but that opening passage does not sound like a call to arms to me. Lucas starts off with an interesting question. He bows respectfully to existing growth theory (including Romer's brand). He notes that they are based on important features of reality and that they are interesting theories. It's just that to him...[now, I want you to brace yourself and to please forgive the man...he is, after all, just an academic trying to explore alternative interpretations of the way the world works] see, to him, the existing theories of growth are not central (i.e., they are missing something that Lucas thinks is more important).

Lucas then proceeds in this highly provocative way: "In this paper I introduce and partially develop a new model of technical change..." From the way DeLong is writing, you'd think that Lucas had instead written "In this paper, I introduce and develop the grand unifying theory of economic growth and development. I await my second Nobel prize."

You can see how much fun I am having with this. Let me continue, along with DeLong, who writes:
The problem comes with the second principal aspect of "mathiness": to claim that one and only one mode of interaction and one and only one mode of individual decision-making is admissible at the foundation level of economic models. Here Romer attacks the assumption that the only allowable interaction is one of price-taking behavior: selling (or buying) as much as one wants at whatever the single fixed price currently offered by the market is. And here I would attack the assumption that individual decision-making is always characterized by rational expectations.
While there are certainly economists who make liberal use of the price-taking assumption in their research, there are equally many who do not. And I am not aware of any economist who would make the claim that the only allowable interaction is one of price-taking behavior. Many economists do feel more strongly about the desirability of imposing "rational expectations." But there is, of course, a vibrant community of those who feel otherwise. And it's not as if people who employ non-rational expectations models are ostracized from the community--unless you call being appointed a central bank president a form ostracism (see, for example, this piece by Jim Bullard).

Why do people have such a bee in their bonnet when it comes to the assumption of price-taking behavior? It's just a pricing protocol (a mechanism that determines prices). Many macro models, especially in the search and matching literature, use bilateral bargaining protocols to determine prices. Monopolistic competition is an assumed market structure together with an assumed pricing protocol. The pricing protocol there is usually quite restrictive: a monopolistic competitor is only permitted to charge a single price for his product. That is, nonlinear pricing schedules (which would increase both profit and social welfare) are assumed away--despite the overwhelming evidence of their use (e.g., retail and wholesale establishments often apply price discounts on large quantities purchased.)

[Aside: Many people are under the impression that monopoly is necessarily inefficient. This is not a valid conclusion. The conclusion follows from an auxiliary assumption that rules out an optimal nonlinear pricing protocol. Walter Oi makes this point in what is surely one of the best titles ever for an economics article: A Disneyland Dilemma: Two-Part Tariffs for a Mickey Mouse Monopoly.]

For Romer, the issue has to do with (I think) of how to reconcile the costly acquisition of nonrivalous (nonexcludable) ideas with perfect competition. DeLong hints at this when he writes:
Thus Paul Romer sees, in growth theory, the current generation of neoclassical economists grind out paper after paper imposing on the world "the restriction of 0 percent excludability of ideas required for [the] Marshallian external increasing returns" necessary for there to even be a price-taking equilibrium.
But DeLong (and Romer) are (I think) wrong on this dimension, at least, on a technical level. It is in fact possible to write down a growth a model where nonrivalous ideas are partially excludable (subject to costly acquisition) in a competitive equilibrium with price-taking behavior. My paper here (with Glenn MacDonald) constitutes one such example.

[Aside: In an extension of my model with Glenn, entrepreneurs are motivated to discover a technological advancement that, conditional on discovery, is assumed to diffuse rapidly among a small but measurable set of firms. This "small" group of firms on the technological frontier is "large" enough to assume price-taking behavior. The technology leaders earn profits (return on their knowledge capital) while laggards attempt to imitate the leaders--an effort that collectively generates the classic S-shaped diffusion dynamic--a phenomenon ignored by most growth models--including Romer's. An alternative and possibly more appealing set up would have been to permit one firm to have a short-lived monopoly right over a technological discovery. I think it's doubtful that any of our main results would have changed under this alternative specification. It's definitely an interesting question to explore and the alternative specification may have implications for questions that we were not interested in pursuing. The point I want to stress is that we did not assume price-taking behavior out of respect for a dogma. It just turned out to be a convenient way to approach things. And I think the approach is still fine, depending on the set of questions the researcher wants to focus on. Telling me that I must behave otherwise in the interest of science seems rather...what's the word I'm looking for here...I think it starts with a D...].

To many people, the assumption of price-taking behavior and rational expectations is thought to imply no useful role for government policy. In fact, nothing could be further from the truth. It is well-known that the equilibria of noncooperative games are generically suboptimal (so that well-designed interventions can be welfare-improving). For that matter, the equilibrium of a monopolistically competitive model can be efficient (and therefore warrant no government intervention). Adopting any one of these assumptions a priori is not, by itself, going to lead to a predetermined policy recommendation. And yet, DeLong seems to suggest that this is indeed the case when he writes:
And I see, in macroeconomics, paper after paper and banker after banker and industrialist after industrialist and technocrat after technocrat and politician after politician claiming that everything that governments might to to speed recovery must be counterproductive, or at least too risky--because that is what is in the case in a very restricted class of rational-expectations models.  
DeLong claims to see (without providing examples), in macroeconomics, "paper after paper" claiming that everything governments do is worse than useless. As evidence for this this, he cites two economists writing in the 1930s. Why not take a look at what people are publishing in high-level journals today? Here is just one example, drawn from the "freshwater" based Journal of Monetary Economics for March 2015. Among the papers published in this issue, I see:

1. Optimal Taxation with Home Production
2. Macroeconomic Regimes
3. Managing Markets with Toxic Assets
4. Financial Stress and Economic Dynamics: The Transmission of Crises
5. Liquidity, Assets and Business Cycles

I'll let this evidence speak for itself. What of his suggestion that the "austerity" forces are motivated by what is true in a restricted class of rational-expectations models? I'm afraid it's just another preposterous statement on his part. As he notes, recommendations of "austerity" were being made even in the 1930s. But that was well before rational-expectations models even existed. I might add that economists today frequently recommend government interventions that rely on rational expectations (e.g., Michael Woodford's "forward guidance" policy proposal).

Let me conclude now by returning to Romer. I'm still not absolutely sure what the mathiness critique is really all about. I think that Romer might just be frustrated (possibly with some justification) that his ideas haven't swept away his competition. Because his ideas are so firmly rooted in logic and evidence, the only thing that can evidently explain the continued resistance is academic politics shrouded in a cloak of mathiness.

Well, that's one interpretation and, who knows, maybe there's an element of truth to it. But my preferred interpretation is that what we seem to have here is simply a healthy clash of ideas competing in the marketplace for ideas. Let's not get too frustrated when our preferred (and obviously superior) theory does not sweep away the competition.

Steve Williamson has his own interesting take on the issue here


  1. Could you please turn on “anyone can comment”?

    Here is how:


    1. Can you describe what sort of difficulty you were experiencing when you made that comment? (Note: I moderate the comments, so you have to wait a little while for me to approve them.)

    2. Sure. I do not think I have any of the accounts in the drop down list where it says "comment as". It took me about 45 minutes to sign up for everything and get the comment to post where it said something about moderated. The moderated part is fine.

      If you go over to Stephen Williamson's blog here:

      the "comment as" drop down list at the bottom has Name/URL and Anonymous available. That would help someone else. It could help me in the future because I might forget the login and/or password.

      Now it is trying to say anonymous instead of uiopl. No idea why.

    3. OK, I changed the setting. Try it out and see if it works.

    4. Testing. Thanks!

  2. While I largely agree with what you're saying, particularly regarding Lucas, I do think you engage in a bit of bait and switch.

    two questions: why the focus on Delong's interpretation instead of the original Romer comments. Many people have offered interpretations, including Romer in offering clarification, yet you go straight to the man most engaged in hyperbole. Romer can hardly be resposible for Delong's rant.

    Why do McGratton and Prescot get a free pass? Seems to me their paper was the most egregious example, the one where Romer had a real point (as Vollrath points out) and yet you spend all your space on Lucas.

    1. Adam P, my answers to your questions are:

      [1] It was the DeLong piece that motivated me to write this and I couldn't very well have critiqued DeLong without mentioning Romer. In any case, their criticisms do overlap in their attack of Lucas. I think it is clear in my post that I do not attribute DeLong's hyperbole to Romer.

      [2] I did not mention McGrattan and Prescott for a few reasons. First, Romer (in his clarification post) does not mention them (he goes after Lucas). Second, Romer's complaint about them essentially boils down to "can you please try to write a little better next time, thank you?" Third, the post was already getting too long.

  3. You are critiquing Romer using the very verbal sloppiness he is calling out. That won't work. Here are two specific examples:

    1) Pretending that there is some confusion about Romer's critique of the Lucas "capital" quote when Romer explicitly stated that the problem was one of implicit logical quantification over models. Maybe you really don't understand. But regardless, your critique of Romer is a Strawman (couched in ridicule). To be specific, neither "by itself" nor "necessarily" is a sufficient qualification on the statement. It should instead read "we also know that doing so in the one model presented here does not provide an engine of sustained growth." Lucas's wording leaves the clear impression that he means "any model". It's arrogant and dismissive if intentional, notably sloppy if unintentional.

    2) Romer is attacking the hiding of (implicit) assumptions in language. He is not accusing people of making explicit claims. Yes you are riffing on DeLong's (mis)use of "claim" as well, but using that one exaggeration by a third party to pile on to Romer makes this statement of yours a complete Strawman fallacy as well: "I am not aware of any economist who would make the claim that the only allowable interaction is one of price-taking behavior."

    As to your sarcastic implication that everyone does it ("all the other styles"): Bingo! That's the very problem Romer is pointing to!

    1. Jeff,

      I qualified my post by saying that I'm an outsider to the field of growth theory, so I may very well have missed the salient point Romer is trying to make.

      You say that Lucas' wording leaves a "clear impression" but it does not to me. And I'm not sure how you square "arrogant and dismissive" with what Lucas wrote in the opening paragraph of his paper. Did you read it?

      I think I make clear when I criticize DeLong I am not also criticizing Romer, except for the part where their criticisms overlap (specifically, in respect to the quote from Lucas' paper).

      Finally, I did not intend to imply that "everyone does it." What I meant was that effective communication is difficult and that it's not fair to single out one or two people for their apparent failures along this dimension.

    2. David,


      I was not referencing anything about the subject matter (growth theory), as I have no knowledge there. My comments are purely at the level of the discourse itself (i.e. meta).

      Lucas may have caveated in the opening paragraph (yes I've read it) and abandoned such elsewhere, in particular in some/all logical-conclusion type statements. This is a natural political/rhetorical move, reifying "my opinion" into "the (only) relevant truth" by the end of the thesis. It is not a valid scientific move without actual deductively logical, valid mathematical, and/or empirical proof (in the paper). That's precisely Romer's critique, it seems to me.

      Here again is Lucas's original statement: "we also know that doing so does not by itself provide an engine of sustained growth". There really is no way to read that as anything other than a supposedly universally authoritative statement (we = economists (uncaveated)) about a universal economic limit (we know ... does not, all uncaveated). How do you read it?

      Finally, demanding Romer call out all examples or none, as the price of being "fair", is, well, completely "unfair" (not a good-faith critique, again purely political).

      Thanks again for the feedback.

    3. Jeff,

      I guess I could find negative subtext anywhere, if I look hard enough. You're making this way too complicated. Simplify.

    4. Here is a thought experiment to understand Lucas's paragraph. Suppose that a virus kills every human being on the planet except those below the age of, say, 8 but leaves intact all physical capital and all scientific books and papers. Now suppose that these survivors are somehow able to continue to educate themselves until they become adults but without guidance from parents or teachers. Would they be able to advance the technological frontier at the same rate as previous generations? If we go by Romer's model, the answer would be yes since the years of schooling (usually considered a proxy for human capital) and the stock of public knowledge have not changed. Is Lucas unreasonable in arguing that the answer should instead be no, that the inter-generational transfer of knowledge through human interaction is important, and that to capture this we need overlapping generations rather than infinite-lived agents? Sorry, but I am siding with Lucas on this one. And to me this is a much more important issue than the choice of pricing protocol.

    5. Stephen,

      The simple interpretation of the Lucas statement IS the universal one -- all economists agree you can't get sustained growth in that manner. It's the defenders that are jumping through hoops to say he somehow didn't really mean "all" or "can't". Right? Thus the charge of sloppy (or ignorant) writing.

    6. Yep. Lot of postmodern redefining of what Lucas said. Given that Lucas also claimed in the early naughties that depression prevention is a problem that has been solved I fail to see why anybody takes this guy seriously.
      Except for freshwater Lucas disciplies like Andolfatto, Williamson and Alexandrakis of course. :D

    7. Being grouped with Williamson and Andolfatto is an honor, which unfortunately I don't deserve.

  4. I think Peter Boettke's thoughts on this discourse are wise:

  5. Andolfatto: "Monopolistic competition is an assumed market structure together with an assumed pricing protocol. The pricing protocol there is usually quite restrictive: a monopolistic competitor is only permitted to charge a single price for his product. That is, nonlinear pricing schedules... are assumed away--despite the overwhelming evidence of their use..."

    As l learned it, the "price taking" models that Romer criticizes is also known as the "perfect competition" model. This is also a single-price model, just one where there are so many firms competing that none of them has any control over the price, so each firm faces a horizontal demand and marginal revenue curve. In the monopolistic competition model, differentiation allows each firm to create mini-markets, and thus downward-sloping demand and marginal revenue curves where each individual firm can increase price by lowering the quantity produced.

    The price taking model is the less realistic one with more unrealistic assumptions. Most advertising that firms do is for the purpose of differentiation, listing features and citing reviews and statistics about what makes their product better than the rest.

  6. I think Romer is right, although his argument may not be. The dynamic equilibrium model is patently absurd. The Firm's "problem" is to maximize output less wages and capital from now until infinity? Why even bring up such a stupid idea? That the consumer maximizes his discounted consumption over infinity? And the first order conditions?

    What does this have to do with anything, and who cares?

    First order conditions are simply meaningless.

    These trvial phrases are simply the result of mathematical functions. They tell us nothing about economics. Further, these models are not really testable in the classical econometric sense - they are simply parametrized to match the data we mine.

  7. Ok, I've now had a good look at your 1998 paper with MacDonald. Do correct me if I'm wrong (I don't grok every detail of the math), but it appears that none of the time that individuals spend in "learning" is paid for by firms. Individuals maximize their ability to earn more wage AFTER learning, but everywhere I see a w_t wage applied, the learning time L_t is subtracted out. True?

    If so, you're smuggling the value of non-rival ideas into the production activity at zero cost (to firms), despite your "costly acquisition" claim above.

    Again, I could well be wrong. If so, can you point me to the math in the paper that accounts for the cost to the firm (of the learning)? Thanks.

    1. Jeff,

      Whether the learning is paid for by the firm or worker is immaterial. The point is that learning is costly. That's what prevents ideas from diffusing instantaneously. I don't think we ever use the words rival and non-rival in the paper. But I recall having in mind the ideas as non-rival, yet embodied in people. I hope that this whole debate isn't about semantics! In any case, I plan to post something that should help clarify. Thanks for taking the time to read the paper.

    2. David,


      But if the monetary price of learning is not accounted for in the model (I believe you folks call that "treated as an externality"?), then you are not modeling the aggregate economy as claimed. It would not be "just semantics", it would be an outright equivocation on "costly" (between the model and your characterization of the model; firms are free-riding on the education system; "costly" as in lost-productivity-only doesn't capture it).

      It could also explain why the trend of the wage rate in the model is higher than the trend in the wage rate in the data (figure 10). (Money that is going to education in the real world is instead accounted as increased wages in the model.)