Paul Romer created a bit of a firestorm over the last week or so with his paper and posts regarding "Mathiness in the Theory of Economic Growth". I finally was able to sit down and think harder about his piece (and several reactions to it).

Before I get to the substance, let me make two caveats. First, Romer has been relentless in continuing to publish blog posts and tweets about his paper, so I'm kind of hopelessly behind at this point. I will probably make some points that someone has made in response, or talk about something that Romer has already brought up elsewhere. If so, the lack of links or attribution is not intentional. I just haven't caught up yet. (See DeLong, and Wren-Lewis for two responses).

Second, one of the papers that Romer discusses is by Bob Lucas and Ben Moll. I know Ben a little, as he gave a talk at UH last year. We recently e-mailed regarding Romer's criticisms of their paper. Some of what I will write below is based off of notes that Ben is writing up as a response. That isn't to say that this post is a defense of Lucas and Moll, just a disclaimer.

What's the issue here? Romer says in his paper:

For the last two decades, growth theory has made no scientific pogress toward a consensus. The challenge is how to model the scale effects introduced by non-rival ideas... To accommodate them, many growth theorists have embraced monpolistic competition, but an influential group of traditionalists continues to support price taking with external increasing returns. The question posed here is why the methods of science have failed to resolve the disagreement between these two groups.

One thing we have come to a consensus on is that economic growth is driven by innovation, and not simply accumulating physical or human capital. That innovation, though, involves non-rival ideas. Non-rival ideas (e.g. calculus) can be used by anyone without limiting anyone else's use of the idea. But modeling a non-rival idea is weird for standard economics, which was built around things that are rival (e.g. a hot dog). In particular, allowing for non-rival ideas in production means that we have increasing returns to scale (if you double all physical inputs *and* the number of ideas then you get more than twice as much output). But if we have increasing returns to scale, why don't we see growth rates accelerating over time? We should be well on our way to infinite output by now with IRS.

To answer that, Romer pioneered the use of monopolistic competition in describing ideas. Basically, even though ideas are non-rival, they *are* excludable. Give someone market power over an idea (i.e. a patent) and this allows them to earn profits on that idea. Because not everyone can actually use the idea for free, this keeps the growth rate from exploding. The profits that an owner of an idea earns from people using it are what incent people to come up with more ideas. So these market power models explain why the IRS doesn't result in infinite output, and explains why people would bother to innovate in the first place.

An alternative is that ideas are non-rival, and non-excludable. That is, anyone is capable of adopting them immediately and for free. To keep the economy from exploding to infinite output, in these models you have to build in some friction to the free flow of ideas between people. Yes, you can freely adopt any idea you find, but it takes you a while to find the new ideas lying around. What you can retain in models like this is the idea of price-taking competition. No market power is necessary for any agent in the model.

Romer's paper then proposes that the lack of consensus on this is due to one side (the latter, price-taking increasing returns group) making arguments for their side not on the basis of scientific evidence, but on `mathiness'. Let's hold off for a moment on that term.

Is this really a big disagreement? In one sense, yes. You certainly still have papers from both camps in top journals, by top economists. In another sense, no. When it comes to doing any kind of empirical work in growth, there is no question that firm-level, market-power models of innovation that grew out of Romer's work are the standard. The problem with the price-taking models is that they say nothing about firm dynamics (e.g. entry and exit), and these dynamics are a huge part of growth. With price-taking, there isn't a reason for any specific firm to exist, and so things like entry and exit aren't well-defined.

Are there math mistakes? The latter part of Romer's paper discusses how several recent examples of price-taking models are sloppy in connecting words and math, and how some of them in fact contain mathematical errors. He discusses, in particular, the Lucas and Moll paper and an issue taking a double limit. This is something that Ben broached with me, and his explanation of the issue seems reasonable, in the sense that Lucas and Moll do not seem wrong. But between Romer, Lucas, Moll, and me, I am the last person you should ask about this.

More important, from the perspective of this `mathiness' question, the math mistakes themselves are irrelevant. Romer's larger point would be worth discussing even if the math were perfect. Pointing out a flaw doesn't change his argument, and in fact probably detracts from it. He isn't asking Lucas and Moll (or the others) to simply correct their paper, he wants to change the way they think about doing research.

Doesn't everyone make silly assumptions? This was Noah Smith's initial reaction to the mathiness post. The assumptions made by the market power theories are just as impossible to justify as the competition theory. The price-taking theory assumes that people just randomly walk around, bump into each other, and magically new ideas spring into existence. The market power theory assumes that people wander into a lab, and then magically new ideas just spring into existence, perhaps arriving in a Poisson-distributed process to make the math easier. Why is the magical arrival of ideas in the lab less fanciful than the magical arrival of ideas from people meeting each other? In the models, they are both governed by arbitrary statistical processes that bear no resemblance to how research actually works.

At their heart, both of these theories have some kind of arbitrary process involved. But that is not Romer's point. Every theory is going to make some kind of fanciful abstraction regarding the real world. If it didn't, it wouldn't be a model, it would be reality.

Okay, smart guy. What is Romer's point? I think it is this: math is not science.

Here's how the science on this would work. Collect data on the growth rate, number of innovations produced, and/or productivity growth. Test whether countries/states/firms that operate with price-taking grow at the same rate as those that operate with market power over ideas. If they do grow at the same rate, then you fail to reject the price-taking theory of innovation. You don't accept it, you fail to reject it. And then you go on your way scrounging for more data to see if that particular test was just a result of sample noise.

If the price-taking market doesn't grow or produce innovations, then you reject the price-taking theory. And then you go on your way scrounging for more data to see if that particular test was just a result of sample noise.

Let's say that you fail to reject the price-taking theory. Now what? Now you start pulling out other predictions from both theories, preferably ones where they differ. And you test those predictions. If you could reject the market power theory predictions, but fail to reject the price-taking predictions, then you'd probably conclude that price-taking is the better explanation of innovative activity (but new data could overturn that). And vice versa. I'm not saying this is easy (how do you identify which economies are price-taking versus market power?), but this is how you'd do it.

That's the science. That doesn't mean the math is worthless. You have to have the math - the model - in order to come up with the predictions and hypotheses that we're going to test with the data. Without the model we don't know how to interpret what we see. Without the model, we don't know what tests to run. So a paper like Lucas and Moll is useful in allowing us distinguish what a price-taking world might look like compared to a market power world.

Here is where we reach the crux of Romer's argument, to me. Most people, including many inside academia who should know better, assume that math equals science. And rather than remind readers that math is not equal to science, authors often play along with that fiction. They play along by using very complicated math - "mathiness" - making their idea look more "science-ish". They let people believe their model shows how the world does work, rather than how it might work.

[Update 5/21 6:30pm: That's not a specific indictment of Lucas/Moll, but a re-statment of Romer's argument. And a valid question here is whether we should expect every paper to actively re-state the concept that models are about how the world might work, not how it does work. Moreover, if someone misuses a theory paper, is that the fault of the authors?]

So all these guys are liars? No, I don't think that Lucas and Moll, for example, are part of some conspiracy. I know Ben is somewhat flummoxed by what their paper did to come under such fire from Romer. They wrote a theoretical paper that strung out the implications of a certain set of assumptions. I think they are perfectly amenable to, and would support, anyone who could come up with good empirical evidence on their model.

How other people use these theories is a different story. Brad DeLong has suggested that the problem with the "mathiness" of these papers is that they allow people to reverse engineer support for their preferred political position. If we have a price-taking competitive economy, then any interference (i.e. taxes or subsidies) will generate deadweight loss. Are we in a price-taking economy? These papers by smart economists show that we *could* be, and if you confuse math with science then you assert that we *are* in a price-taking economy. Hence no interference is justified.

Why can't we all just get along? Perhaps we should have different models for different situations. Dani Rodrik has made this point before (H/T Israel Arroyo for the link), urging economists to focus on choosing the right model, not trying to shove everything into one grand unified theory. The market power theory is useful in understanding innovation in pharmaceuticals, for example, or innovation in a leading-edge Western country like the U.S.

But the price-taking theory is useful in a situation where the innovation we are talking about is not actually a brand new idea, but rather an existing idea (even an old one) that people have not adopted yet. Think of something like proper fertilizer application among peasant farmers. Some farmers use the fertilizer properly, some don't. But this isn't because some farmers have property rights over the knowledge of how to use fertilizer. How long it takes the good practices to diffuse over the whole population of farmers may well be modeled as a a series of interactions between farmers over time, and knowledge gets passed along at each step. Taking the arrival of truly new innovations as exogenous may be a reasonable assumption to make for some developing countries.

If Lucas and Moll had framed their theory this way, would that mitigate the mathiness of the paper? I think it might.

Now what? I think Romer's paper is right that we are not careful enough about distinguishing math from science in economics. It is easy to slip, and I have no doubt that some people take advantage of this slippage to push their viewpoints.

One thing to insist on (of papers you referee, of speakers, or of one's own work) is that falsifiable predictions are clearly stated. What could the data show that would make your theory wrong? Force the authors to be clear on how science can be used to evaluate your theory. That isn't to say that every theoretical paper needs to have an empirical test added to it. I am always in favor of smaller, more concise papers. But the follow-up empirical work for your theory should be obvious. Then hope some grad student doesn't take the bait and prove you wrong.