In his comment on my Mathiness paper, Noah Smith asks for more evidence that the theory in the McGrattan-Prescott paper that I cite is any worse than the theory I compare it to by Robert Solow and Gary Becker. I agree with Brad DeLong’s defense of the Solow model. I’ll elaborate, by using the familiar analogy that theory is to the world as a map is to terrain.

There is no such thing as the perfect map. This does not mean that the incoherent scribbling of McGrattan and Prescott are on a par with the coherent, low-resolution Solow map that is so simple that all economists have memorized it. Nor with the Becker map that has become part of the everyday mental model of people inside and outside of economics.

Noah also notes that I go into more detail about the problems in the Lucas and Moll (2014) paper. Just to be clear, this is not because it is worse than the papers by McGrattan and Prescott or Boldrin and Levine. Honestly, I’d be hard pressed to say which is the worst. They all display the sloppy mixture of words and symbols that I’m calling mathiness. Each is awful in its own special way.

What should worry economists is the pattern, not any one of these papers. And our response. Why do we seem resigned to tolerating papers like this? What cumulative harm are they doing?

The resignation is why I conjectured that we are stuck in a lemons equilibrium in the market for mathematical theory. Noah’s jaded question–Is the theory of McGrattan-Prescott really any worse than the theory of Solow and Becker?–may be indicative of what many economists feel after years of being bullied by bad theory. And as I note in the paper, this resignation may be why empirically minded economists like Piketty and Zucman stay as far away from theory as possible.

Perhaps looking back at good theory will serve as a partial antidote.

The Solow model is an example of excellent theory. It is a remarkably compact and approachable general equilibrium model. It was the perfect retort to the claim by Stigler and Friedman that Marshall gave economists all the tools that they would ever need.

Brad is right to emphasize that it made a prediction about the effect of changes in saving rates, but remember that it also predicts the behavior of the real wage, the real interest rate, and labor’s share. On the empirical side, it facilitated a deep dive into questions about measurement that other economists, most prominently Dale Jorgenson and Zvi Griliches pursued. (Sure, Dale and Zvi were brash about how much better their measures were than Solow’s, but they were clearly building on his work.) The results that emerged from this work are now reflected in our national system of statistics.

Like any good model, the Solow model focused on some questions and set others aside for later. My work on endogenous technological change was an extension of the theory just as the work of Jorgenson and Griliches was an extension of Solow’s empirical work. (Sure, I was brash about claiming how different the extension was from to the original model, but I was clearly building on his work.)

Becker’s analysis of human capital is another example of excellent theory. Becker did something that even Solow did not do. He took a new phrase and added it to the vocabulary of people outside of economics. If you were not around, you may not realize how deeply offended many people, even economists, were by his promotion of the phrase (which, it must be said, already had a long but sporadic history of use in economics.) Initially, it must have seemed totally implausible that it could ever achieve nearly universal acceptance.

Becker did for human capital what Newton did for the term force. It is rare intellectual accomplishment. These two cases illustrate perfectly how mathematical theory differs from mathiness. In these cases, mathematical theory gave a precise definition for a word or phrase. This precision made it so useful that it has become part of our everyday vocabulary.

The existing word “capital” was probably forced on Solow by history. It still carries along many of its non-technical meanings (e.g. the working capital of a firm.) Solow’s math sharpened how economists use the term and forced us to be more careful about the meaning of the derived concept of investment. But outside of economics, hardly anyone understands these words to mean what economists think they means.

Becker succeeded inside economics partly because his theory had immediate implications for measurement and evidence that spanned micro and macro. And, to be sure, because he could build on the work of Jacob Mincer. Without the theory, we could still have run all kinds of micro regressions, but we would not have been able to link the micro evidence with the aggregate theory.

If you think that what McGrattan and Prescott do for location is even remotely on the same level as what Solow did for capital or what Becker did for human capital, please go read the two M-P papers (JET 2009, AER 2010.)

When you think you are too stupid to understand what they are saying and want to give up, trust me, it isn’t you. What they are saying makes no sense. No one can understand it. The authors do not understand it.

Here is a sample of what you can expect:

Technology capital is distinguished from other types of capital in that a firm can use it simultaneously in multiple domestic and foreign locations. (Footnote: In the language of classical general equilibrium theory, a unit of technology capital is a set of technologies, with one technology for each location.) (JET 2009, p. 2455)

“A unit … is a set”? This is just gibberish. Forget about whether the model connects in any meaningful way to the real world. There is no way to make sense of this statement even in the make-believe world of the model. In the model, the authors define technology capital is a cardinal measure. It is supposed to be something that you can have 2 units of, or 4, or 10. What could 2 or 4 or 10 sets of technologies possibly mean?

We assume that the measure of a country’s production locations is proportional to its population, since locations correspond to markets and some measure of people defines a market. (JET 2009 p. 2461)

I feel guilty pulling a quote like this one, as if I’m humiliating some miserable undergraduate by reading to the class from a term paper on a project that fell apart. But remember, this is from an article that was published in the Journal of Economic Theory.

As you read this quote, remember that the motivation for the theory is that for these authors, perfect competition is the ultimate non-negotiable, more sacred even than micro-foundations. If this were a Hotelling model of location or a Krugman model of spatial location, I’d have some way to try to make sense about how “some measure of people defines a market.” But in the formal mathematical model of perfect competition that the authors are using, this sentence means nothing.

These words are untethered, undisciplined by logic or math, chosen to sound plausible enough to someone who is not paying close attention, like the set up for an applause line in a speech by a politician. This is mathiness.

There is lots more:

One unit of technology capital and z units of the composite input at a given location produce y = g(z). Consider the case of brand equity with units of technology capital indexed by m. For ease of exposition, assume for now that m is discrete and that m = 1 is the Wal-Mart brand, m = 2 is the Home Depot brand, and so on. Wal-Mart chooses the locations in which to set up stores and use its brand. It may be the case that both Wal-Mart and Home Depot have stores at the same location. (AER 2010, p. 1497.)

And if you look at the math, a company like Wal-Mart has to use one unit of technology capital for each location. Because the number of locations in the US is the US population, Wal-Mart must be using more than 300 million units of technology capital. (So more than 300 million technology sets?)

How can we reconcile the math with words that say Wal-Mart gets index m=1 and Home Depot gets m=2? And if technology capital is brand equity, why does Wal-Mart need another unit of brand equity for each US citizen/location? I haven’t a clue, but neither do the authors. One of the things that Milton Friedman got right was his observation that “confused writing is a sign of confused thinking.”

As a discussant, I put serious effort into trying to clean up the mess in the working paper that became the 2009 JET paper. I worked through the math. I talked with the authors.

The things I explained, such as how to convert any concave function like g(z) into a function with one additional variable that is homogeneous of degree one, just helped them put lipstick on this pig.

It was an embarrassment for me that the 2007 NBER version contained the acknowledgement “Discussions with Robert Lucas and Paul Romer were extremely helpful…”

Here’s a good way to test whether anyone thinks that the theory in the McGrattan-Prescott papers is on a par with the theory developed by Becker or Solow. I’ll put up $10,000. They can tell me what odds they would have to get to take the other side of a bet where I win if the McGrattan-Prescott notion of location does not lead to a revision of the official US statistical system.

Same for a bet that their concept of location not achieve wide acceptance outside of economics. Or even in economics.

Noah notes that the Solow model wasn’t very explicit about the residual, but good theory always tees up follow-on work. I unpacked the residual by exploring technology as a nonrival good. Klenow and Hsieh went in another direction, revisiting questions about aggregation and reallocation.

Paul Krugman’s chapter “The Rise and Fall of Development Economics,” from Development, Geography, and Economic Theory, gives a clear eyed description of the costs and benefits of mathematical theory that makes the case for why theory should be part of our tool kit. His is my favorite exposition of the map-terrain analogy, but it has a long history and there are many other versions.

For specific purposes, some maps are better than others. Sometimes a subway map is better than a topographical map. Sometimes it is the other way around. Starting with any good map, we can always increase the resolution and add detail.

No map is perfect, but this does not mean that all maps are equal. It certainly does not mean that an internally consistent map that with so little detail that you can memorize it is on a par with incoherent scribbling.