The Bohr model of the atom, showing an electron transitioning from one orbit to another by emitting a photon. CREDIT: Wikipedia CC BY-SA 3.0

Paul Romer mis-handles atomic physics

He thinks that some of the core mathematical models of modern economics are every bit as sound and scientific as Niels Bohr’s 1913 model of the atom. It’s not really true.


Economist Paul Romer recently attacked some of his colleagues for employing what he called “mathiness” in their research. Nobel Prize winners Robert Lucas and Edward Prescott were among his targets. Romer claimed that these economists, in constructing theories about economic growth, have often been guided mostly by ideology and politics. They’ve slipped preposterous assumptions into their economic models to guarantee the results they want, and hide their subterfuge behind clouds of complicated-looking mathematics. Their work is mostly politics, and has little to do with science.

Romer’s attack ignited a fierce debate to which many people have contributed, including myself, Noah Smith and Brad DeLong. I suggested that much of the mathiness of modern economics has come about because economists, for historical reasons, use mathematics like mathematicians — following the influence of Gerard Debreu and the famous Bourbaki school. They don’t use it in the practical way that physicists and engineers use it.

Feeling that I (and others) had misunderstood his views, Romer has since tried to clarify his thinking in several further posts. In one in particular, he suggests that he actually thinks the formal Debreu style of building models — starting with axioms, and deriving propositions through the use of logic — is quite all right. Moreover, he believes that models such as the original Arrow-Debreu model of a competitive equilibrium work much like their counterparts do in physics, and should be held in high esteem for this reason. He compared the Arrow-Debreu model to the Bohr model of the atom, introduced by Neils Bohr in 1913, in an effort to explain how atoms absorb and emit light only at specific frequencies. As Romer put it,

There are a few instances in the development of a discipline in which pure Bohr-style science (which is not motivated by any specific problem) can yield a big return. I’d cite the work that Arrow, Debreu, and McKenzie did on formalizing a general equilibrium as an example of successful Bohr-style science that focused on tool building.

When I first read this I assumed that Romer, with the term “Bohr-like,” must have been referring to some peculiar philosophical position put forth by Bohr. He couldn’t, I thought, actually be making an analogy between Arrow-Debreu (add McKenzie if you want) and Bohr’s model of the atom. That seemed doubly impossible given his use of the phrase “not motivated by any specific problem,” given that Bohr’s model was very much motivated by a specific problem. But after further reading, it seems to me that this is precisely what Romer meant — that these two theories, one in physics and one in economics, work in a parallel logical fashion. And, hence, that this kind of theoretical work is an example of economics at its best.

I think this is completely confused, so let me explain why. The only way, it seems to me, that Romer can link these two theories in his head is if he has completely forgotten why Bohr proposed his model, and what they model achieved.

Physicists in 1913 knew that the hydrogen atom — the simplest atom, with one electron orbiting around a proton — absorbed or emitted light at a particular, discrete set of frequencies. This was hugely puzzling, because in classical physics radiation gets generated by the acceleration of charged particles, and the frequencies produced reflect the nature of that acceleration. Heat hydrogen, and you should expect that, as atoms bang into one another, electrons ought to be driven to undergo all manner of complex motions, certainly not motions just at one or a few discrete frequencies. You would expect their motion to create a broad and continuous spectrum of radiation. Why on earth do atoms, in reality, radiate at only a few specific frequencies?

This is the extremely specific and practical problem that Bohr had in mind. Why specific frequencies only, and why the particular frequencies actually observed? To attempt some explanation, Bohr made a bold supposition — unsupported by anything in earlier physics — that perhaps electrons can only inhabit orbits having specific values of angular momentum, with those allowed being integral multiples of the new fundamental quantum of action, h, introduced earlier by Max Planck. This would imply that an atom would have a series of states of distinct energies. Bohr then added to this a second, even bolder supposition — that the frequency of radiation given off by an atom as an electron goes from a higher state to a lower one is simply the energy difference between those two states divided by Planck’s constant h. This is weird because it says that the frequency of the radiation coming out has nothing to do with the frequency of electron motion in either the beginning or ending state. In essence, it dismisses electron motion entirely in working out the frequencies of radiation.

From any classical point of view, this is crazy. Bohr dreamed it up only as a desperate measure to find some way to account for experimental observations. In contrast to Romer’s claim, Bohr was completely motivated by a very specific problem. And everyone would have ignored the theory entirely had it not been spectacularly successful in explaining just what is observed. When Bohr worked out the energy levels for a hydrogen atom, and then the frequencies of light generated by an electron falling from a higher to a lower state, he found precisely the frequencies that experimenters observed from heated hydrogen gases. You have one series of lines, the Lyman series, for transitions in which electrons end up in the lowest energy state, another series, the Balmer series, if the electron ends up in the 2nd lowest state, and so on. Bohr’s theory made a few simple if strange assumptions, and had a spectacular empirical success in explaining a broad set of data. The Bohr model soon failed, when applied to more complex multi-electron atoms, but it was a great leap forward for a time.

Now compare this to the Arrow-Debreu theory of competitive equilibrium. As far as I’m aware, this model was indeed not motivated by any specific problem. Rather, it was designed to give a mathematical form to economists’ prior idea — going back ultimately to Adam Smith — that prices might help a market to match the aims of producers and consumers in an efficient way. Arrow and Debreu showed in a highly abstract model, with a host of assumptions, that there would exist a unique set of prices that would achieve this. The only similarity to the Bohr model is that Arrow and Debreu stated some assumptions and derived some implications. There the similarity ends.

The analogy would only hold if it were the case that the Arrow-Debreu theory made immediate and specific quantitative predictions for a host of things that could be measured, and that these predictions turned out to be just right in many cases. This, I believe, is very much not the case. I don’t think that economists ever invoke the Arrow-Debreu work as a direct explanation of some set of economic data. It is, rather, just a formal statement of a picture of the world that many economists like to hold in their minds. They like it as a mathematical object, not as a theory of anything real.

In this, the Arrow-Debreu model shares quite a lot with most of the models used by economists. They’re models of the Debreu-Bourbaki style — formal mathematical structures not intended to be directly compared to reality. Economics, for many economists, including Romer it seems, is the exploration of mathematical models of the kind that economists like to study, some of which are of the kind initially proposed by Arrow and Debreu.

So, in the end, I would agree with Noah Smith that mathiness is all over economics, and Romer’s criticism applies to the field at large, not just to some specific works of Lucas and Prescott. As Smith notes, the basic approach that economic theorists use….

… would never fly in engineering. Engineering is something you expect to work. But macroeconomists often treat their models as simply ways, in the words of David Andolfatto, vice president of the Federal Reserve Bank of St. Louis, to “organize our thinking” about the world. In other words, macroeconomists use math to make their thoughts concrete, to persuade others, and to check the internal consistency of their (sometimes preposterous) ideas, but not to actually predict things in the real world.
Back to Romer’s complaint. He singles out Lucas, Prescott and a few others for having tenuous or sloppy links between mathematical elements and the real world. But from what I can see, such tenuous and sloppy links are the rule in macro fields. Romer says that I am “jaded” for saying that, and that it was bad apples like Lucas and Prescott who soured me on macroeconomic theory. Well, he’s right that I’m jaded, and he’s right that it was learning about models by Prescott that I first became jaded. Romer, you got me.
But when I looked beyond those models, to older or newer models, I found what seemed to be only a difference in degree, not in kind. Macroeconomic theory is chock full of mathiness. It’s not just Lucas and Prescott, it’s the whole scientific culture of the field.

Exactly. Bohr’s model is almost the anti-thesis of theorizing in the style of Arrow-Debreu, or Lucas, or Prescott — or Romer.