Krugman: I’m For Math!

Krugman clarifies:

I’ve been getting some comments from people who think my magazine piece was an attack on the use of mathematics in economics. It wasn’t…So by all means let’s have math in economics — but as our servant, not our master.

Word.

(Of course the point I was trying to make was that I read the end of his article as suggesting that because economics must deal with the irrational and unpredictable behavior of humans, that it must therefor be messy and beyond elegant mathematical description. I don’t buy this line of reasoning, as I think it is unknown whether the conclusion is true, but apparently, reading comments to my article, I’m the only one who doesn’t like to put his mathematics before his solution π )

But anyway, is anyone going to explain inflation without using gauge theory? (Channeling Eric Weinstein)

1. #1 Victor Riley
September 11, 2009

Let me suggest another blog post (http://economistsview.typepad.com/economistsview/2009/09/tacit-knowledge.html) that might bring this together, in which Mark Thoma cites another post by Daniel Little about tacit knowledge (knowledge that is not formalized or propositional). Thoma makes the point that, when he was taught econometrics, his teacher emphasized the tacit side of modeling, which recognizes that models are only formal and therefore limited. Putting these concepts together suggests that the problem with math in economics is the failure to recognize its limits, and the key to using it properly is to combine it with human insight, perspective, and the ability to assess its relevance, accuracy, and limitations in light of the bigger context that the model can’t capture.

2. #2 John Sidles
September 11, 2009

Dave, isn’t gauge invariance strictly a computational convenience for weak-minded folks who resort to (oh the shame of it) coordinate systems and quotient manifolds?

“Mathematicians of Mana” prefer to contemplate state-spaces—both classical and quantum—as pure, coordinate-free, geometric forms!

Of course, this makes it mighty tough to actually calculate anything …

… but fortunately, if we simply prove that a problem has a solution … that’s good enough! π

3. #3 David
September 11, 2009

“Mathematicians of Mana” prefer to contemplate state-spaces—both classical and quantum—as pure, coordinate-free, geometric forms!

Of course, this makes it mighty tough to actually calculate anything …

… but fortunately, if we simply prove that a problem has a solution … that’s good enough! π

Sometimes I think it’s easier to learn about category theory from non-category-theorists. This is one of them.

4. #4 Eric Weinstein
September 11, 2009

Dave,

Given the level of the confusion above about Fiber Bundles, perhaps a few words are in order. Here comes the worlds shortest intro to Gauge Theory:

Gauge theory is about doing differential calculus where the analog of the x-y plane is something even more interesting.

The x-axis is given freedom to be a manifold, and the y-axis and it’s translates are given freedom to become diffeomorphic to Lie Groups (e.g. think of a donut if you need a mental placeholder). The x-y plane is allowed to be a twisted combination of these modified x and y axes. Then as an analog of the x-y plane you try to do calculus on this thing called a fiber bundle….except that you can’t do “rise over run” without some some reference from which to measure the rise. That reference is called a connection or gauge field or covariant derivative. The interesting thing here is that there are many such connections so you need structure to pick relevant connections.

That connection is an honest first order differential operator but with the property that it’s mixed partials don’t commute as in calculus. The obstruction to commuting is a generalization of familiar geometric curvature which instead of being called meta-curvature is just called curvature.

This is more or less what we believe lies behind all four forces, so sit up straight ’cause “weak minded” it ain’t. I think this is arguably the most powerful thing in Mathematics. Think of it as a classical gateway drug to
quantum field theory.

Our point is that at the heart of economics lies a massive fiber bundle with connection. It’s a bet that the entire field is blind to the fact that it sits atop a gold mine. One connection that does something new and the whole field could change. It’s not for sissys.

Keep up the great work Dave,

Eric

P.S. Remember to keep asking for the cost of living definition under taste change. It’s the best way to reassure yourself something freakishly weird is going on here. Otherwise I just think I’ve gone nuts.

5. #5 John Sidles
September 12, 2009

Eric, what I liked about your response was that it never mentioned the word “invariance”!

Because “gauge invariance” is not really what gauge theories are all about, is it? In the same sense that “diffeomorphism” is not really what GR is all about, is it?

Terry Tao has a terrific post on these issues, titled What is a Gauge? whose concluding paragraph is:

“It would also be desirable to have a more gauge-invariant theory of PDEs that did not rely so heavily on gauge theory at all, but this seems to be rather difficult; many of our most powerful tools in PDE (for instance, the Fourier transform) are highly non-gauge-invariant, which makes it very inconvenient to try to analyse these equations in a purely gauge-invariant setting.”

Now, if gauge theories are not about gauge invariance, then what are they about? For me, they’re all about the interplay between the symplectic structures that govern Hamiltonian dynamical processes, versus the metric structures that govern measurement and noise processes … each structure alone is simple … its when we mix them that the spukhafte Fernwirkung of quantum mechanics emerges.

A skiing metaphor is that metric structure tells us how to ski downhill … symplectic structure tells us how to ski sidehill … and when both structures are present, hey, we’re ready for some free-style skiing!

6. #6 Rocky Humbert
September 12, 2009

Here’s John Cochrane’s (the UChicago prof, not OJ’s lawyer) response, entitled, “How did Paul Krugman get it so Wrong?”

http://faculty.chicagobooth.edu/john.cochrane/research/Papers/krugman_response.doc

7. #7 John Sidles
September 12, 2009

Uggghhhhh … Cochrane’s response is in Microssofttttt Woooorrrrdddddd … (flesh crawls & zombies start beating on the window).

Anyone wanna bet there are no equations in it … that Cochrane’s main goal is to adjust, not the practice of economics, but our expections of economics? π

8. #8 John Sidles
September 12, 2009

Well … having read Cochrane’s article … it’s utterly horrible … there’s no math in it whatsoever … seemingly because Cochrane betrays an utter lack of understanding of complexity theory, and of the intimate relation of complexity to to the moral foundations of modern economic theory.

Yes, I said “moral foundations of modern economic theory” … those moral foundations being, the preservation of the moral values that have long been associated (since the time of Spinoza, Locke, and Jefferson) with individual enterprise.

Why is individual enterprise so important? And what distinguishes individual enterprise from corporate enterprise?

For any complexity theorist the answer is both simple and obvious: individuals have vastly lower computational capacity than corporations.

The point being, as soon as markets evolved—under the moral false-flag of efficiency!—a level of complexity incompatible with individual enterprise … such that economic activity quenched rather than nurtured individual enterprise … then the moral foundations for modern economics were wholly compromised.

Prof. Cochrane has a reputation for being intelligent … it was surprising to discern in his essay no understanding whatsoever of these (elementary!) mathematical and historical principles.

———

THERE! Was that review “snarky” enough for me to qualify as a Chicago-school economist? π π π

9. #9 Tyler DiPietro
September 13, 2009

Cochrane’s review is difficult to take seriously when it contains lines of reasoning like this:

“Itβs fun to say we didnβt see the crisis coming, but the central empirical prediction of the efficient markets hypothesis is precisely that nobody can tell where markets are going β neither benevolent government bureaucrats, nor crafty hedge-fund managers, nor ivory-tower academics. This is probably the best-tested proposition in all the social sciences. Krugman knows this, so all he can do is huff and puff about his dislike for a theory whose central prediction is that nobody can be a reliable soothsayer. “

Um…so, if no one can realistically predict the behavior of markets, what the hell is economics supposed to study? If it is content to base a theory (efficient markets) on a gigantic argument from ignorance (no one can predict the behavior, therefore we’re right) then Krugman isn’t the one who should be compared t a creationist.

10. #10 John Sidles
September 13, 2009

— snarky Cochrane review, Part II —

It is striking that Cochrane’s economic world-view is utterly lacking in significant predictive power. Indeed, his economics is (in a sense) designed to lack predictive power.

For example, it is striking that one year into a global financial collapse—a year during which the urgent need for regulatory reform has been universally acknowledged—no substantive reforms have yet been implemented.

This is striking economic behavior—even amazing behavior!—that surely, any good system of economics needs to predict and explain. Here Cochrane’s framework fails utterly … yet amazingly, Cochrane himself seems utterly oblivious to this failing.

The reason is, of course, that Cochrane-style economics has become an ideology rather than a social science … an ideology whose mainspring is the protection a world of corporate-dominated economic behavior, and the academic “bubble” that is subsidized by this corporate world.

—————————————

Gosh … writing snarky economic reviews of (lets face it) weak essays like Cochrane’s is lots easier and (in a shameful way) more fun than pulling back KΓ€hler triples and getting all the signs and factors of two right in the Lie derivatives of symplectic and metric tensors! π

11. #11 Rocky Humbert
September 13, 2009

JS: et tu Professor?

You wrote, “Cochrane’s economic world-view is utterly lacking in significant predictive power.” I challenge YOU to find a single practicing economist of any philosophy, who, over a lifetime, has demonstrated any statistically significant ability to profitably PREDICT market prices.

You wrote: “a year during which the urgent need for regulatory reform has been universally acknowledged”. I can unequivocably state that your use of the term “universal” is factually wrong. (This is the sort of grandiose Krugman-esque rhetoric that is not based in fact.) What IS universally obvious is that existing regulations and laws were not enforced either intentionally or due to regulator incompetence. (e.g. Cochrane points to Madoff.) There is an important distinct between ENFORCEMENT of existing laws, and “reform” (which means different things to different people.)

As for post-hoc prediction, I suggest you read the following paper:

http://www.stanford.edu/~johntayl/FCPR.pdf

Cheers, (and before you make an ad hominem attack against me, I reserve the right to change my mind, since I am none of a polemic, partisan, politician or a great thinker [such as you, are.]

12. #12 John Sidles
September 13, 2009

With respect, Rocky, the ability to … “demonstrate any statistically significant ability to profitably PREDICT market prices” … is definitely *not* (IMHO) what economics is all about!

Because isn’t that like saying … “the ability to predict the course of a disease is what medicine is all about” … which is a too-limited conception of the profession of medicine! π

If you want to say “Cochrane-style economics is indubitably correct … except insofar as it is practiced by people who are short-sighted, incompetent, greedy, and/or corrupt” … well heck … I’m happy to stipulate to that … even Krugman would be happy to stipulate to that! π

13. #13 Rocky Humbert
September 13, 2009

Tyler writes: “Um…so, if no one can realistically predict the behavior of markets, what the hell is economics supposed to study? If it is content to base a theory (efficient markets) on a gigantic argument from ignorance (no one can predict the behavior, therefore we’re right) then Krugman isn’t the one who should be compared t a creationist”

There’s a lot more to economics than predict market behavior. Heck, as a market participant for almost 3 decades, I can demonstrate that ANY asset price can swing +/-20% based entirely on changing investor preference.

I suggest you dig up your old Econ 101 text book on the chapter discussing supply, demand and equilbrium. Unless the chapter has been revised since I took the class decades ago, basic microeconomic theory included assumptions such as frictionless transactions, perfect information, perfect competition etc etc. None of these conditions exist in practice, but that does not render the theory useless.

As for rational behavior (and my comment about +/- 20%) — there’s always a way to show that someone’s behavior is rational. The problem here is that because one participant’s rational decision may appear to be another participant’s irrational decision, it becomes complex to correctly model all of these different utility/preference curves. This is no way diminishes the virtue of research into these problems. Krugman knows this. Cochrane knows this. And I suspect, you know this too.

14. #14 John Sidles
September 13, 2009

As a followup, if Cochrane’s essay had included a well-conceived section titled: “On the economic motivations for behaviors that are short-sighted, incompetent, greedy, and/or corrupt” … well … IMHO Cochrane’s essay have been *greatly* improved … in particular with regard to his framework’s predictive power! π

15. #15 Tyler DiPietro
September 13, 2009

“There’s a lot more to economics than predict market behavior. Heck, as a market participant for almost 3 decades, I can demonstrate that ANY asset price can swing +/-20% based entirely on changing investor preference.”

And there is a lot more to predicting market behavior than predicting short term asset prices. Economics could, for instance, make claims about long term regularities. But regardless of how you slice it, Cochrane is making an argument from ignorance: the lack of an empirical observation proves me right; because markets are unpredictable, my model is right.

“As for rational behavior (and my comment about +/- 20%) — there’s always a way to show that someone’s behavior is rational.”

Yes there is, by expanding the definition of “rational” to include anything that could potentially be observed. That may be clever sophistry, but it sure as hell isn’t science.

16. #16 Rocky Humbert
September 13, 2009

Professor JS:

I would humbly submit that Cochrane’s core weltanschauung takes account of actors whose behavioral motivations include time-horizon and greed (aka profit maximization.)

As for incompetence, free markets eventually penalize this characteristic (which is one reason why GM, Citicorp etc found themselves in their current pickle.) [You mght choose to argue that the social costs of this incompetence are too great for society to bear, (which is why regulation is necessary), but you haven’t done so.] And, this argument which you chose not to make, could be usefully modeled and tested using exactly the mathematics that Krugman scorns.

But that brings us to “corruption.” Cochrane sees no a priori reason to believe that the government should be any less corrupt than private actors — and in fact, there are good reasons to believe that they may be more corrupt (ceteris paribus). I think this is a key point where Krugman gets into trouble.

Lastly, whereas incompetence in private actors results in bankruptcy, incompetence in the government results in calls among the Krugmans for larger government budgets, and more power to that part of the system, which has failed us the most. This is one of the more salient points that Cochrane makes, and is exactly the current state of affairs.

I miss Milton Friedman. He would have written a much better response.

17. #17 John Sidles
September 13, 2009

Rocky Humbert: it becomes complex to correctly model all of these different utility/preference curves

Gosh … isn’t that a pretty considerable understatement? Don’t sociopaths, crooks, con-artists, ideologues, religious fanatics, and addicts of every kind, all have “utility/preference curves”?

And yet, who would want to participate in a marketplace that offered sanctuary—or even rewards!—to these people?

A major problem with Cochrane’s “pure” marketplace framework is that it offered shelter—and even immense rewards—to precisely these people.

Isn’t that right?

It is significant (IMHO) that concepts like “empathy” play no role in Cochrane’s economic framework.

Recent cognitive research has shown the people vary widely in their aptitude and capacity for empathy. Perhaps Cochrane—and his econometric colleagues—are not particularly gifted in this area?

18. #18 Rocky Humbert
September 13, 2009

Professor JS:

“Don’t sociopaths, crooks, con-artists, ideologues, religious fanatics, and addicts of every kind, all have “utility/preference curves”? ”

Yes they do. So do illegal immigrants, union bosses, newspaper columnists, and liberal academics who claim to have superior knowledge and wisdom (despite their never having started a business, hired employees and paid wages to their employees, some of which turned out to be sociopaths, crooks, con-artists, ideologues, religious fanatics and addicts.)

“A major problem with Cochrane’s “pure” marketplace framework is that it offered shelter—and even immense rewards—to precisely these people.”

Cochrane is offering huge rewards to sociopaths, crooks, con-artists, ideologues, religious fanatics and addicts? I must have missed that part of his essay. Can you direct me there please. If so, I stand corrected.

Empathy is a lame (almost religious) justification for opinions without quantification and rigorous analysis; (Another Cochrane point, by the way.)

Here’s a laughably trite example of the challenges of using “empathy” absent a rational cost/benefit analysis:

Highway deaths are horrible, expensive and often needless.
Highway guard rails save lifes. They prevent drivers from going off cliffs. They prevent head on collisions. Guard rails can be engineered to stop a car dead cold (ouch), or they can be engineered to absorb energy and be deformed by impact (more expensive).

I can direct you to countless, rigorous studies that show we could save X lives and Y catastrophic injuries if we would only spend Z dollars to install and maintain guardrails on every paved road in America.

But am I lacking empathy to argue that society needs to weigh the costs versus the benefits before imposing this sort of “regulation” on the entire country (even though it will save lives)? Do you have any empathy for the employee/entrepreneur who can’t afford to buy something that he genuinely wants to buy, because he is being forced to pay higher taxes so noone drives off the road?

If one’s family member (G_d forbid) goes off a road that doesn’t have a guard rail, that is a tragedy. But do we, as a society, lack empathy because we recognize that the societal costs of “guard rails for all” can not justify the societal benefits?

Admittedly, this is a trite example. But it illustrates how quantitative analysis is the least worst approach to decision making.

Ultimately, I’d argue, that economics is the study of the different ways to allocate/share/divide resources and the consequences of those decisions. That has nothing to do with empathy, or the lack of it.

p.s. I’ve run out of words. So if you want the last word, be my guest and godspeed. Thanks for sharing your thoughts.

Rocky

19. #19 John Sidles
September 13, 2009

Rocky, it’s tough to reply when you start with a false imputation “Cochrane is offering huge rewards to sociopaths, crooks, con-artists, ideologues, religious fanatics and addicts?”

Definitely never said it! Because there’s a big difference between “offering rewards” (which I didn’t say) and “offering shelter” (which I *did* say).

Purely on the facts, is there any doubt that, over the last decade, our system offered too much shelter to “sociopaths, crooks, con-artists, ideologues, religious fanatics and addicts?”

And I will go further, to assert that Cochrane-style economic policy (very unwisely) offered “huge rewards” to those sectors of the economy whose complexity exceeded the capacity of any individual to understand those sectors.

These policies was implemented under the disastrously improvident ideological false-flag of “market efficiency” … which (in the event) turned out to create a protected habitat for players who turned out to be (who else?) sociopaths, crooks, con-artists, and ideologues.

The resulting global financial melt-down is the Waterloo of Cochrane-style economics … and this I take to be Krugman’s (entirely valid) point.

I’d be sorry not to hear more from you, Rocky, because your arguments have offered some nicely instructive points for rebuttal! π

20. #20 Rocky Humbert
September 13, 2009

Oops. I forgot to write in my farewell to alms that your final riposte will likely include:
1) No direct rebuttal to my points one by one.
2) A cynical and condescending tone.
3) Gross generalizations without any quantifiable statements that cannot be proved or disproved.
4) Use of hyperbole, such as “Waterloo,” as well as suggesting false imputations/connotations … when they are there for all to see.
4) A failure to acknowledge where one’s bread is buttered (e.g. perhaps as a academic union employee whose labors and pension are involuntarily funded by taxpayers?) or perhaps as an entrepreneur/capitalist who risks his capital every day in the pursuit of profit, the creation of jobs, and a better standard of living for all, and who when successful in his endeavors sees nearly 60% of his income paid to the professor who mocks him.

The beauty of democracy and the internet is readers of this blog can reach their own conclusions…. The beauty of markets is that you know when you were right and when you are wrong … and the beauty of tenure is …. ????

21. #21 Tyler DiPietro
September 13, 2009

Shorter Rocky: “How dare you mock me? I pay your salary!”

22. #22 Rock Humber
September 13, 2009

Shortest Rocky: “and that government of the people, by the people, for the people, shall not perish from the earth.”

23. #23 Tyler DiPietro
September 13, 2009

lolwut

24. #24 John Sidles
September 13, 2009

OK, here’s your report card, Rocky …

(1) “No point-by-point rebuttal”? Happy to oblige!

(2) “Cynicism” … by no means! … “condescending” … OK, half-credit! π

(3) “Quantifiable statements” … hasn’t Madoff already been convicted by the peer review of a jury, and sentenced to 150 years in jail? That’s pretty “quantifiable” , isn’t it? And surely, aren’t many more “sociopaths, crooks, con-artists” destined to follow in Madoff’s footsteps? π

(4) “Hyperbole” … hmmm … after all, didn’t Napoleon salvage something from Waterloo? He retreated successfully to Paris, didn’t he? Where he still retained hopes of victory? So perhaps my little essay instead deserves criticism for understatement? π

(5) Your email scores about 20% with regard to its public guesses about the private lives of me and my family … that family life (which is of course our private business!)—although interesting to the point of being picaresque—encompasses a panoply of activities and passions that do not conform to any academic or economic stereotype … as I hope it always will! π

Because that is the human condition, isn’t it? π

25. #25 Rocky Humbert
September 14, 2009

Professor JS:

I suspect that you will agree that there is fine line between mutually respectful discussion/disagreement, and disrespectful mocking, ad hominem attacks, and irrelevant slights. To the extent that I committed any vituperous abuse in this thread, I apologize. I also apologize out of respect for Dave Bacon, whose intellect is remarkable and whose blog is most interesting, (perhaps more so prior to my contributions.) Mud-wrestling, while an entertaining pursuit for a autumn Sunday afternoon, is degrading to us all.

I read a study recently that showed debate regarding these sorts of issues rarely changes one’s views; and rather causes more polemicism. This exchange might be an example of that. So, I’m going back to what I do best (which isn’t debating), you should go back to what you do best (which probably has little to do with defending Krugman)…and the world will be a better place if we both succeed in our pursuits.

26. #26 John Sidles
September 14, 2009

Rocky, your last post was your best, and it will be mighty tough for me to match it! Surely, it is not necessary that everyone think alike (it would not even be desirable). Your closing sentiment “the world will be a better place if we both succeed in our pursuits” was IMHO wholly admirable. The best of luck to you, and to all your pursuits, and to everyone you care about!