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« Why we use math in economics | Main | Why do good economists publish in the WSJ editorial page? »

September 05, 2007

Comments

Per Kurowski

Re: And call me naive, but I also think that Mugabe would not have pursued his policies for this long if he had a better grasp of debt dynamics.

Sorry I cannot think of any debt dynamics model that could handle a swan as black as Mugabe.

Ken Houghton

I'm all for expertise, but when what we get as passing for Economics Expertise With Math is Dick Cheney's Napkin, I'm inclined to think that cactus's summary (http://angrybear.blogspot.com/2007/09/economics-is-bull.html) is correct:

In the end... the reason economics is bull$%!# is because the practitioners have allowed it to remain bull$%!#.

If Economics Professors are not Rational Actors, then we have to assume no one is. So let us assume they are RAs and construct a model that explains why data is eschewed and The Napkin is invoked.

The rest is a bit of algebra, some regression analysis, and a few touches of higher math--none of which will make phlogiston into gold.

James

As someone who wants to be an economist and is currently struggling with Calc III, I was very heartened by your comment, "Neither of the applications I mentioned above requires rocket science or even calculus beyond what is taught in a good high school." Great post and keep up the good work!

robertdfeinman

I haven't read too many theoretical economics papers, but those that I have read seem to be swatting a fly with a sledgehammer.

Perhaps someone can explain to me why anything more than normal statistical tools is required for any economic model which claims to based upon observable data?

Economics, after all, claims to be a social science and thus is trying to generalize human activity into a simplified model.

If the observed behaviors really have as many variables as, say, weather forecasting, then perhaps this is direction that the mathematics should go.

If that turns out to be the case, then the use of probabilistic models and stochastic techniques should be examined. Computers are now powerful enough to deal with thousands of inputs. If you have every seen the models of tornado dynamics you know what I mean.

The fundamental problem that remains, it seems to me, is sparse data. Just look at the debates over what the Fed should do next. Most of what it appears they are using as input is aggregate data such as changes in the GDP or employment rate, or the like.

In a multi-sector economy, like ours, this seems too abstract. We could be having a boom in one area while another is contracting. A single policy that would address both areas optimally would seem to be impossible.

Bruce Wilder

Let me quote without any further embellishment from a comment by "Matt" left on a Brad DeLong essay:

"The Solow growth model assumes labor devoted to consumption and labor devoted to capital stock are decorrelated. I know this because his exponentials are real valued, so the covariance matrix is not hermitian, not cyclic. The two eigen functions are orthogonal. But there is no valid real kernel approximation to such a long period of time. This also explains the Malthusian trap."

inthemachine

Just to clarify, I meant the original challenge for Terence, who seemed to suggest that math and development failure were somehow related. Of course math is necessary for economics - and for making sound economic decisions - how else are you going to consider and communicate about n-dimensional issues.

Apologies that my sarcasm clouded the point that when development programs or projects fail, it is never because "there was math" or because "there was not math," which seemed to be the direction some of the previous commets were going.

krishna

I also think math has tremendous utility and power in clarifying and making accurate a theory (like Robert Feinman, I am a physicist). But one problem I had when reading economic theory papers is that there is a tremendous amount of effort spent on making their mathematical models *rigorous*, i.e in proving them in a mathematically strong fashion (this is true even of the most famous economic theorists, like Lucas). It makes little sense to me to do this, because the models themselves are so imperfect, and often inconsistent with data. Would it not be more reasonable to spend this effort on fixing inconsistencies between the models and empirical data, rather than proving their mathematical exactness?I see very in my profession spend their time proving their models rigorously.

Justin Rietz

First, I think it is well known by readers of this blog and mine that I have a slight distaste for econometric modeling. Let me make it clear that I believe the use of math for teaching economic principals (calculus for marginal calculations), clarifying and explaining ideas, and doing sound statistical analysis of real world data is important and useful.

Some of my knocks against complex econometric models are:

1. Cleansing of data by removing outliers. While it is sometimes justified, it may just well be that these outliers are important data points that reflect an inconsistency in the theoretical assumptions underlying the model. I think there is a great danger that economists, unintentionally, shape data to fit preconceived notions of what they expect to find.

2. Similar to #1, the large role that assumptions play in determining what is useful and what is not. Again, unintentional biases by modelers may have a significant impact on such assumptions.

2. The inability to accurately capture critical human elements dependent upon politics, culture, and other social factors. There have been some interesting studies about this phenomena - I'll see if I can dig up some links.

3. The lack of attention given to correlation vs. causation found in many econometric-based research papers

4. The substituting of econometric models for on-the-ground research by economists who prefer to stay in their offices and theorize about people and events happening someplace thousands of miles away that they have never visited (just to be clear, I do not put Dani in this category).

I'm reading "Inventing Money" by Nicholas Dunbar, a book about the failure of Long Term Capital Management that focuses on the theory of LTCM's financial models (Black-Scholes, etc.) I have found some interesting correlations with our current topic. To put it a bit simplistically: theoretical, mathematical models can be quickly blown away by real world events.

Peter

"In macroeconomic policy, understanding and knowing how to work with the fundamental debt dynamics equation is critical to the conduct of fiscal and monetary policies."

Funny you should bring this up, since I just published a piece in Challenge that argues that most of the economics profession, despite its high-tech pretentions, is culpable of one of the most elementary mathematical errors, confusing identity and functional relations. The national accounting identities cannot be invoked to "explain" why causation supposedly runs from savings behavior to the current account.

When I reviewed the writings of prominent economists on this topic in preparation for the article, I was awestruck by the extent of the confusion.

Perhaps the right conclusion to draw is that math proficiency is, as you say, very important, but that real understanding is what matters most, rather than oodles of technique. (But, yes, there are some problems where technique is crucial.)

hari

Dani-

Economics is NOT rocket science, and must deal with social issues - more often than not - which cannot be readily quantified (without creating a static model!).

Of course, most economic texts become more redable with graphs,tables and whatnots.

The problem with model building in (global) political economy - unlike in physical sciences - one has to deal with macroeconomics of nations with indifferent statistical standards.

Let me take a real-life example to illustrate this issue:

When EU decided to expand and include (former) East European Communist States, EUROSTAT issued a red signal! Eurostat had to develop a statistical dossier for each applicant, and found it simply couldn't produce a standard document for EU negotiators. History will record EU (political) negotiations didn't begin until Eurostat had succeeded in developing a statistical standard for new applicants. Even then, it took a lot of time before the system was officially formalized.

Therefore, economists must always be aware that Chinese and Indian statisical data are not the same as official data issued by OECD for its member countries.

Shampa

Perhaps the question is not whether math should or shouldn't be used in development economics but how much of the development story it actually tells us? Development stories are generally complex, nuanced and case-specific and similarly figuring out what works and doesn't in development doesn't usually lend itself to just a pure mathematical interpretation. Balancing the information obtained from mathematical models with rigorous qualitative data assists in unpacking the black box. The danger is that once a model spits out a number, unenlightened policy-makers could in some cases latch onto it to make bad simplistic "magic bullet"-type decisions rather than delving into the shades of grey that qualitative data might reveal.

Barkley Rosser

There are really two different types of math going on here, and it is probably useful to keep them separate. One is the math used in pure theory and the other is the math used in econometric analysis of data. Of course, nobody likes to try to read a paper that uses math they do not understand, but sometimes such math is necessary for the task at hand.

The usual vice of the theorist is to make unrealistic assumptions and then take them too seriously as reflecting reality, irrespective of the level or type of math used in building the theoretical model and analysis. Clearly, Dani has lots of sympathy with this critique, and is more in favor of empirical analysis anyway.

Regarding empirics, well, problems with data can lead one to use more complicated econometric techniques to overcome these problems. That said, such techniques are more useful in disproving something that appears to be true from a simpler analysis (usually just good old OLS). However, I remain convinced that if it does not show up in the data pretty straightforwardly, indeed in the old eyeball test, it probably is not really there. And even if it is there via a fancy technique, if it does not pass the eyeball test, you are never going to convince a policymaker that it is there, especially one who is not convinced already that it is and wants it to be.

Matt Nolan

From what I understand, economists used to love doing economics with no maths.

Then someone discovered that maths was a neater way of illustrating some concepts. This discovery illustrated that some of arguments people were making were inconsequential, as mathematically they were saying the same thing.

The criticism of words at this point was that people used words to hide there inability to illustrate the phenomenon with a model.

In the last couple of decades the models have become extremely technical, and many economists will make assumptions based on the fact they are common in the literature. The small nuances that could so easily be described with words are shown through numerical models, and some of the economists that derive these models do not seem to understand the full extent of the environment they are describing. So in some sense, economists now use maths to hide the fact that they don't understand what they are doing.

Anyone that can describe there model clearly and succinctly without maths is a good economist.

terence

Oh - I feel obliged to say that I certainly didn't mean to sound derisive. Just light-hearted.

I do think there are two serious points that can be made in critique of the original post:

1. That math alone won't save you from bad economics (hence my jibe about the 1980s). (But I doubt this is news to anyone).

2. (More substantively) There is a lot to be said for mathematical rigor (and Dani's post makes an excellent case for this) but if you take this too far you run the risk of lapsing into the realms of extreme positivism, where things that can't be counted (or modeled) don't exist and where maths becomes the only means of understanding the world. Certainly without the formal logic of mathematics you do run the risk of woolly thinking but if you restrict yourself only to this means of understanding you seriously limit the things you may understand.

Anyhow, the most important point here is that I really enjoy reading this blog (it is my morning highlight) and certainly wouldn't ever want to deride its author.

notsneaky

"Perhaps someone can explain to me why anything more than normal statistical tools is required for any economic model which claims to based upon observable data?"

Perhaps I could if you make your question more precise. What are "normal" statistical tools? Are fancy econometric techniques - essentially extensions of the simple linear regression to particular data and statistical problems - "non normal"?

Or do you mean that economist should just use statistical techniques and forget about constructing mathematical theoretical models, which, uh, ultimately underlie those "normal" statistical techniques one is gonna use?

"As someone who wants to be an economist and is currently struggling with Calc III, I was very heartened by your comment, "Neither of the applications I mentioned above requires rocket science or even calculus beyond what is taught in a good high school.""

Unfortunately, in my experience it seems often to be the case that in order to truly understand a particular set of tools - to the point where one is able to use them and apply them outside the settings in which one learned them - one needs to study stuff that is two or three levels removed. So if you just wanna roughly understand what the papers in AER, JPE or QJE are talking about you can probably get on fine with Calc I. Well, Calc II. But if you wanna build your own models and *really* understand the stuff at the deeper level, you DO need the higher maths (but it's fine to get them from econ courses). Or have a lot of practice.

"The problem with model building in (global) political economy - unlike in physical sciences - one has to deal with macroeconomics of nations with indifferent statistical standards."

Yes and a lot of time and energy is expended by economists in trying to make the available data comparable (sometimes this is called "harmonization"). See for example Robert Feenstra's work with trade data or the Penn World Tables for that matter.

"the economics profession, despite its high-tech pretentions, is culpable of one of the most elementary mathematical errors, confusing identity and functional relations. The national accounting identities cannot be invoked to "explain" why causation supposedly runs from savings behavior to the current account."

Uhhh, this is well known. That's the difference between an identity and theory. Identity just tells you that when one variable changes another one's gotta give. The theory tries to tell you (perhaps unsuccessfully) which ones and maybe how much. So while S=I+X is an identity the statement that when there is an exogenous fall in savings X will fall by dX is a theory.

"When I reviewed the writings of prominent economists on this topic in preparation for the article, I was awestruck by the extent of the confusion."

You got examples or a link to your article? Somehow I'm suspicious of your claim.

"a comment by "Matt" left on a Brad DeLong essay"

Umm, I take this as an example of "too much math". It is a good example, since obviously the guy has no idea of what he's talking about. I'm willing to bet serious money he's not an economist. Maybe an over eager beginning student of economics.

Ok, I'll stop now.

pat toche

I have just read the two posts on mathematics, and I think this second post and the comments are well to the point. I just left a post there which attempts to answer what I think is the key problem to be addressed, as nicely put by Robert Feinman, why do so many economics papers "swat a fly with a sledgehammer."

The answer, I think, is that many economists seek to make their models as general as possible in the belief that if an idea is theoretically robust to the topology used to frame it, it will follow that the idea may be more relevant to the real world. But this need not be so, and the real test of an idea in economics is to put it to the data. As Dani correctly points out, this is increasingly being done. The smartest economists today do applied economics. The profession is changing dramatically, I think.

But it will take a few years before we can see the impact of this transformation. We are still under the declining influence of the 1960s and 1970s, where economists sought to check the theoretical validity of their ideas. We are now moving towards empirical economics. The most innovative economists these days do empirical stuff, the likes of Barro, Mankiw, Heckman, Card, Krueger, Levitt, etc.

So economics is changing, but it will take time for the change to be more clearly visible.

pat toche

To Terence,
"if you take this too far you run the risk of lapsing into the realms of extreme positivism, where things that can't be counted (or modeled) don't exist and where maths becomes the only means of understanding the world"

I have just learned that I am an "extreme positivist".

I also agree with the related idea articulated by Wittgenstein, "Whereof one cannot speak, thereof must one be silent"

viking

Whether in economics and other sciences, soft or hard, math is simply a tool to ensure that an argument or proposition is internally consistent, all assumptions and hypothesis are explicit, and therefore that a position opposed to another can be properly distinguished. Without elementary math and logic, tractable debates on how the world works or doesn't are impossible.

robertdfeinman

notsneaky:
"Normal statistical techniques" are those taught in courses on statistics.

There has been some interesting work recently in public health with meta studies. This is where the results of a large number of prior experiments or observations are then treated to further analysis. This is how it is possible to extract small effects from a large amount of uncorrelated data. It is what is frequently used to find side effects in drugs that weren't noticed during trials. Look up the Vioxx story if you want to see the technique at work. If enough people start to use these tools, they will also become "normal".

One of the dirty secrets of much economic and sociological work is that those conducting the studies have an ideological agenda. This throws their entire work into question, so they try to obscure things with fancy mathematics.

The day that a "study" comes out of Cato which proves that free-trade is a bad idea, let me know...

Barkley Rosser

I would also like to point out that having equations does not necessarily equal "math" and not having equations does not necessarily mean "not math." Math involves the ideas and their nature.

Thus, John Chipman has argued (sorry, no cite) that John Stuart Mill invented nonlinear programming in his chapter on exchange rates in one of the later editions of his Principles of Political Economy. He did so strictly using words in a verbal description. But, there is no question, he laid out a rigorous (and correct) mathematical argument. Also, while she generally said bad things about using math(s), Joan Robinson's growth theory models were generally mathematically rigorous, even if usually described strictly in words, although she would sometimes resort to figures or graphs, which is getting a bit more overtly mathy.

OTOH, there is a lot of fake or lousy math out there in econ that is not really math, that is really obfuscation and sometimes even wrong, irrelevant to what it is supposedly modeling or explaining, and so and so forth. There are equations, but the variables in them are meaningless, and there is no empirical content. They may even be made to look fancier and more complicated than they are with, unnecessary changes in variables, and so forth. Such things are not really math, even though they may look like it (and I am especially aware of this today, as I just finished writing a negative referee report on a paper of exactly this sort, with incoherent variables and even outright incorrect math).

robertdfeinman

I'll ask the professional economists, especially those who teach, if there is lots of shoddy work being done in economics papers as Barkley Rosser seems to think, then why?

Are students taught mathematical obfuscation explicitly? Are there schools of mathematical thought so that one technique is used by one school and another group does something else?

If the techniques are explicitly taught then they must be felt to be valid by those doing the teaching. Are there debates within the profession about this?

It reminds me of the fights at the beginning of the 20th Century between the Freudians and all the opposing camps. That wasn't science either.

corvad


"To Terence,
"if you take this too far you run the risk of lapsing into the realms of extreme positivism, where things that can't be counted (or modeled) don't exist and where maths becomes the only means of understanding the world"

I have just learned that I am an "extreme positivist".

I also agree with the related idea articulated by Wittgenstein, "Whereof one cannot speak, thereof must one be silent""

I believe Terence was making the point that quantification of the real is

a real narrow approach to describing reality and that Wittgenstein was making the point that (here, out of context pastiche style, surely but pastiched with Terence and pattoche) indeed, saying nothing is preferable to extreme positivism.

The Randomizer

Over the past decade or so, the math pre-reqs for getting into most leading programs in economics have ballooned, to the extent that it is (or at least was, until a couple of years) very difficult to get in without at least a math minor (incl. topology, real analysis etc.) As a result, the whole world of professional economics - founded on the concept of simplifying assumptions to describe generalities about social behavior - has been turned on its head and is now centered on complexifying assumptions and the manufacture of esoteric theoretical constructs of scant applicability to anything.

I've been asking economists for a long time to explain the necessity of the level of mathematization that now consumes the discipline and have found few coherent answers. The unfortunate conclusion I have come to is that math functions as a guild system for economists. Without the math, the discipline would be a lot more accessible to lay people (or at least those lay people armed with a desire to read books) and the perceived value of the professional economist (and thus the graduate program in economics) would thereby be a lot less.

This is not to say there is no justification for some math in economics. Of course, there is, but to standardize and simplify. The level of math that currently exists in the discipline renders what are simple concepts when described in textual language, more esoteric and complex. This adds negative value to the pursuit of scientific truth.

Until a few years ago, the state of economics was on a very grave trajectory. With the fall of barriers faced by international students wishing to pursue graduate in the U.S., graduate admissions became a lot more competitive during the 1990s. The response of economics departments, at least, was to select increasingly on the quality of mathematical training (as noted above). This result was pools of graduate students more interested in doing theoretical math than actual economics. That their work had little practical value was not of their concern, since their motivation was to construct mathematical beauty rather than to seek empirical truths.

Thankfully, the situation was saved somewhat by the exogenous shock that was the Poverty Action Lab (developments in matching techniques helped too), which reintroduced economics to science. How durable the impact of that shock will be against the incentives which produced the guild system of hyper-mathematization will be interesting to see . . .

paine

qunatitive thinking
may only require commercial arithmetic

give market prices are rational numbers

compare
facility
with a few
soft ware programs
with learning
to crack
a few diffy q
nuts for control theory target practice

( my old mentor vivian walsh
had a nice piece somewhere
on edgeworth math
as the glass slipper of economics )

rational speculation
has its appeal
and not just to logicians with a fondness for proper acting nymphets

and as noticed above
if tenure track research
makes it possible
it will bloom

( example
from an earlier era
too much time
spent on refining
altogether too abstract qualitative results
like unique equilibrium existence proofs
such runic gibberish
merely
mimics math
far more then
it applies it to economics
call it minor bird math
by math minors )

and not only
is it very often the wrong math
its also
a paotlatch contest
a serious time and tuition waster
too

analogy

the fashion
of
writing journal articles
in latin
especially after
say 1550

was it a lingua franca
or using a useless ornament of genteel education??
ie
a covert
gauntlet /barrier
to entering hoi- hois

some of both no doubt

but subsequent science
published in native tongues
shows it was primarily
a mandarin system

similarly( as pointed out
by ram above )
high math hurdles
exist as much for their own sake

topology ???

why even complex functions ???


either may well
winnow the field
to certain brain types
more or less useful
but the skills specifically acquired....

well
its as if
fencing lessons and horse riding
were required at west point

notsneaky

"This result was pools of graduate students more interested in doing theoretical math than actual economics. "

I really really think this is incorrect if you're talking about 1990's and after. As Dani notes the trend in economics has been towards empirical work and away from abstract theory (personally I actually think the pendulum has swung too far the other way).

Also, there really is a need to be more specific about which "fancy maths" people are talking about.
Is it the fancy econometric techniques? Or is it theoretical models which at this point may be too abstract to allow for empirical analysis?

Mark Thoma

Alfred Marshall on Mathematics in Economics

...In 1906 Alfred Marshall wrote about his skepticism regarding the use of mathematics in economics[1]: [I had] a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules -
(1) Use mathematics as a shorthand language, rather than an engine of inquiry.
(2) Keep to them till you have done.
(3) Translate into English.
(4) Then illustrate by examples that are important in real life.
(5) Burn the mathematics. (6) If you can't succeed in (4), burn (3). This last I did often.

http://economistsview.typepad.com/economistsview/2006/05/alfred_marshall.html

Will Bond

A great contributor to this topic is Krugman - check out "Models and Metafores", the third chapter of his pamphlet "Development, Geography, and Economic Theory".

Krugman makes a strong case for the value of models, and the mathematics which support these models.

Darren McHugh

Dani, I note that your examples of useful technique in economics require nothing more sophisticated than algebra (OK, maybe a bit of first year calculus too). I agree that this is the "sweet spot", but I point out that the level of mathematical sophistication expected of entering graduate students greatly exceeds this minimum threshold. There absolutely IS a crowding out effect... students who can eat a great deal of math for breakfast are rewarded, those who would prefer their economics to be 30% math and 70% history, demography, institutions, methodology (ie the RIGHT mix) are chased out.

Notsneaky claims that this has changed, but my own experience with a top tier terminal master's program in 2005 in Canada was 100% abstract mathematics, and looking back I consider the experience to have been, for the most part, a spectacular and sickening waste of time and tuition that has left me with a pronounced lack of respect for academia.

Darren McHugh

Barkley Rosser: "There are equations, but the variables in them are meaningless, and there is no empirical content. They may even be made to look fancier and more complicated than they are with unnecessary changes in variables, and so forth."

Barkley has just perfectly described my Master's program.

derrida derider

On the swing of the pendulum back towards data-, rather than theory-, driven economics, and the reasons for that swing, it's worth reading Angus Deaton at http://tinyurl.com/34ud2q

bob

For me this issue is all about Universities seeking to improve the perceived quality of their economics degree courses because by adding more math you add more rigour.

This becomes problematic when more math becomes a (very poor in my view) substitute for enabling students to develop a strong conceptual grasp of economics.

An example:
In my Undergrad economics course we spent hours and hours learning the math for complicated models.

But how long did we spend on two of the most brilliant and powerful economic models - the prisoners dilemma and the market for lemons - about half an hour each.

In my humble view, the end result of all this is that some of the brightest people to study economics quickly make the decision to do something far more relevant.

Andrew N

I will attempt to illustrate my opinion of the hyper-mathematization of economics with analogy:

Theoretical econometric models are like an aeronautical engineer trying to utilize the physics of aerodynamics to predict the flight of a bird. While it is true that the path of their flight must follow certain laws of physics, a biologist familiar with the species will likely be able to better predict the bird's course than the engineer.

To take this analogy a bit further (hopefully not too far), one might think of price theory's usefulness in predicting the behavior of an economic system as the gravitational constant with regard to the bird's flight. The system becomes exponentially more complex as the variables enter into the equation, such as the wind speed, elevation, behavioral patterns of the species, individual characteristics of the bird, ad infinitum, until a mathematical prediction becomes absurdly impossible.

Many economists, it seems to me, have forgotten their roots in sociology, psychology and political science. They found they had an easily quantifiable data source (i.e. $) with which to apply mathematical rigor to their discipline. While the usefulness of this measure is profound, it is grossly overstated in modern economics.

I have never seen a natural scientist attempt to mathematically predict the behavior of any other species. Any attempt would rightly be met with more than a little skepticism. It is only by shrouding their workings in esoteric mathematical models that economists have gotten away with it for so long.

That said, I believe that the discipline of economics is invaluable because, unlike a bird on the wing, we can deliberately design the system and principles under which our economy operates. At least to the extent that we can predict and control how the members of our society operate, at which point we return to the flight of a bird.

Thorstein Veblen

Let me put in my two cents:

The first: With having a liberal education, once you have it, you know it has value. You don't need to argue. Such is the case with knowledge of mathematics. So I'll skip the debate over whether it's useful.

The 2nd cent: Dani completely skipped the debate about whether math is all graduate students in Economics should do. The first year of Grad school in Econ is critical -- it's where you learn to sweat, and it determines who stays in the program and who is kicked out. It is also generally devoid of readin', writin', and any systematic treatment of any economic concepts or policy at all. Case in point: learning about "social security" only via OLG models. Solving problems, quick, tight algebra, and proof writing/memorization are now the skills Economists consider to be the most important. (Mankiw justifies this truth on his blog since this apparently allows Economists to pin down students IQ...) The more technical your previous degree, the easier time you'll have surviving the first year. So, more mainland Chinese/engineering majors will become economists than, say, undergrad history or political science majors, who wouldn't have a prayer. (Although those w/ strictly technical backgrounds tend to find research difficult, as it's a different skill. Even then, they can just stick to applied math, which is much of the field.)

Also, responding to posters, I don't think it particularly difficult to connect Grad Econ education w/ failed IMF/World Bank policies such as the Washington Consensus, much less large blocks of bullshi!t "academic" research...

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This result was pools of graduate students more interested in doing theoretical math than actual economics.

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