
Kenneth Arrow and the golden age of economic theory - Hooke
http://voxeu.org/article/ideas-kenneth-arrow
======
georgeglue1
To further appreciate what a giant Arrow was, this is a good overview with a
couple of other examples as well: [http://www-
siepr.stanford.edu/ArrowShovenMay09.pdf](http://www-
siepr.stanford.edu/ArrowShovenMay09.pdf)

------
unknown_apostle
« That age – which includes such historically important figures as Arrow’s
fellow Nobel laureates Paul Samuelson and Gary Becker – represented a
development and expansion of formal economic theory that brought unprecedented
precision to the logical foundations of social science. »

And here we are, picking the fruits of this golden age of unprecedented
precision. With a mountain of record low-yielding debt serving as the backbone
for a mountain range of derivatives and synthetic products. With low savings
and with record public participation in investments. Because the infomercials
say the risk^D^D^D^Dvolatility is controlled by science and computers n shit.

------
paulpauper
Sightly off-topic, but IMHO the theory of random walks, risk neutral pricing,
and the black scholes option pricing formula to be the most significant
findings of the past 100 years in the field of economics, in term of:
practical applications (the option and futures market is huge, and they all
involve these formulas), spawning research (if you go on arXiv, the original
work done in the late 60's and 70's is still spawning tons of research to this
day on asset pricing and asset price dynamics, whereas other economic fields
seem to have stagnated), and being mathematically empirically sound (meaning
it's a complete theory that does an adequate job describing reality), and is
was groundbreaking in that the result was unexpected and answered a nagging
question about how to price contracts without having to define a drift
variable.

~~~
srean
> theory of random walks, risk neutral pricing, and the black scholes option
> pricing formula to be the most significant findings

Random walks, heck yes! Black Scholes not so much. Its a fantastic toy model,
but over all I think, it did more harm than good. People took its thin tail
behavior too seriously. There's way too much fluctuation in practice than a
Gaussian process would fit/predict.

So I would put it as: mathematically sound enough, empirically not quite so.

~~~
JumpCrisscross
> _Its a fantastic toy model, but over all I think, it did more harm than
> good. People took its thin tail behavior too seriously._

Options theory is popularly misunderstood. Makes being a former options trader
fun and annoying.

First, the impact. You can model almost anything as an option. Equity? Call
option on a firm's assets struck at the liabilities. (Literally anything else?
A portfolio of Arrow–Debreu options [0].) Modern risk models, and these
include some very successful ones, would not work without Black Scholes.

Second, no professional uses a Gaussian assumption without knowing what
they're doing. (In any case, people started re-writing Black Scholes for other
curves since at least the 1990s.) No model can economically ascertain every
possible risk. One must always choose risks one considers negligible. This is
true in finance as in life. Sometimes we choose to ignore the wrong risks.
When that happens, it's easier to blame highfalutin math than admit "I didn't
think about that".

No amount of "fat tailing" will substitute rigorous risk management. Case in
point: the multitude of funds launched after 2008 focussing on "black swans"
and "fat tails". Pretty much all of them lost money [3]. They missed--with
everyone else--Dubai's near default, Greece's actual default, the 2010 Flash
Crash, 2011's summer volatility, China's 2015 crash, the following August's
international crash, Brexit or practically anything else that one might
consider both significant and unexpected. (No shit.)

The real "magic" in Black Scholes? It's not the distribution. It's volatility.
Modern models expand this once single term into a multidimensional beast [4].
This is the other reason for the prevalence of Gaussian assumptions. Traders
trade computation of the distribution for computation of the volatility
surface. The latter (tau) almost always dominates the former (delta) in terms
of what's being mispriced.

[0]
[https://en.wikipedia.org/wiki/Arrow–Debreu_model](https://en.wikipedia.org/wiki/Arrow–Debreu_model)

[1]
[https://en.wikipedia.org/wiki/Portfolio_insurance](https://en.wikipedia.org/wiki/Portfolio_insurance)

[2]
[https://en.wikipedia.org/wiki/Tulip_mania](https://en.wikipedia.org/wiki/Tulip_mania)

[3] [http://www.businessinsider.com/watch-out-investors-the-
black...](http://www.businessinsider.com/watch-out-investors-the-black-swan-
funds-arent-that-good-2009-6)

[4] [http://edoc.hu-
berlin.de/series/sfb-649-papers/2005-20/PDF/2...](http://edoc.hu-
berlin.de/series/sfb-649-papers/2005-20/PDF/20.pdf)

~~~
srean
Vehemently agree, in particular this

> no professional uses a Gaussian assumption without knowing what they're
> doing

I would say Black Scholes set the theme, but the details had to be modified
quite a bit. Unmodified BS is BS.

'Fat tailing' would be very unlikely to give one a better prediction accuracy.
It can, however, give one a better appreciation of the risk.

~~~
JumpCrisscross
> _Unmodified [Black Scholes] is BS_

Black-Scholes-Merton is a theory. Analogous to Newtonian mechanics or textbook
thermodynamics. They all need modification to work in the real world. That
doesn't make them BS.

> _[Using a fat-tailed distribution] can, however, give one a better
> appreciation of the risk_

How does one choose a model and parameters for events which are, by
definition, hard to predict? Keep in mind that most "black swans" arise from
unforeseen dimensions of risk. There's the "my stock lost 90% of its value"
fat tail versus "the damn exchange went bust". There's "my clearing bank is
broke" and "the Russians invaded my country".

------
graycat
Gee, it's fun to read about Arrow's work!

From the OP, I see that I've been closer to Arrow's work than I knew!

In grad school, I had a relatively severe introduction to optimization
including both linear and non-linear programming.

The non-linear programming was mostly about the Kuhn-Tucker conditions, and
there the work was mostly about the Kuhn-Tucker necessary conditions. Kuhn and
Tucker were long at Princeton. The guy who was the Chair of my Ph.D. oral exam
had been a Tucker student.

Before I got to that grad program, I had carefully studied W. Rudin,
'Principles of Mathematical Analysis' (a.k.a. Baby Rudin) and W. Fleming,
'Functions of Several Variables'. In my first year of grad school, I also had
a severe course in H. Royden, 'Real Analysis', the real part of W. Rudin,
'Real and Complex Analysis', Neveu, Breiman, Chung, etc.

So, that background gave tools that helped attack the Kuhn-Tucker conditions.

Intuitive view of the Kuhn-Tucker conditions (KTC): You are in a cave with an
uneven floor and vertical walls and you want to find the lowest point. If you
put down a marble and it starts to roll, then you are not at the lowest point.
So, to be at the lowest point, it is necessary that the marble not roll.

But K-T wanted more: They wanted to say that necessarily the slope of the
floor (calculus gradient) and the slopes of the constraints that define the
walls are such that the slopes from the walls block moving along the slope of
the floor. Right, the slopes from the walls form a cone that contains the
slope from the floor (or its negative depending on maximizing or minimizing
and the direction of the constraints, etc.) -- it's all about a cone.

Well, this stronger statement is true in _nice_ cases, and for a _nice_ case
have to have some assumptions that the constraints are _nice_. For that there
are various KI _constraint qualifications_ , KTCQ, that are enough to make the
KT statements about slopes true.

There are lots of KT CQs, and one question was, which imply the others?

For two famous KT CQs, one due to KT and one due to Zangwill, it was not known
if they were independent.

So, as a grad student, I settled that -- they are independent. The proof was
by counterexample -- I found some bizarre constraints.

To know that such bizarre (goofy, pathological, etc.) constraints could exist,
I needed essentially a theorem

For a positive integer n, the real numbers R, Euclidean n-space R^n with the
usual topology, and a subset C of R^n closed in that topology, there exists a
function

f: R^n --> R

such that f is zero on closed set C, strictly positive otherwise, and
infinitely differentiable. So, I proved that. For the KT CQ I didn't need all
of infinitely differentiable, but I got that also.

This result is curious in part because some examples of a close set C can be
surprisingly intricate, e.g., the Mandelbrot set, a sample path of Brownian
motion, Cantor sets of positive measure, etc.

As I went to publish, I discovered that my work also answered a question asked
but not answered in a paper by Arrow, Hurwicz, and Uzawa.

Of course Arrow got his Nobel Prize. A few years ago, so did Hurwicz. Last I
heard, Uzawa was still waiting! Cute: As a grad student I answered a question
asked but not answered by Arrow, Hurwicz, and Uzawa. Reading Rudin and Fleming
helped!

Gee, in the OP, I see that Arrow was also interested in decision making under
uncertainty. Well, my dissertation research was in best decision making over
time under uncertainty -- stochastic optimal control.

I never took a course in economics. My Ph.D. advisor thought that I would need
such a course if only later in my career to fend off nonsense objections from
economists -- I've never needed that!

So, I signed up for an econ course, went the first day, sat in the front row,
said nothing, and took careful notes. After the class when just the professor
and I were there, I asked him what he was assuming for his supply and demand
curves \-- continuous, uniformly continuous, differentiable, continuously
differentiable, infinitely differentiable, convex, pseudo-convex, quasi-
convex, etc.? He said nothing.

Soon I got a call from my department secretary to call my Ph.D. advisor -- I
was out of the econ course!

Still, the OP shows that I was closer to some mathematical economics than I
knew!

Maybe someday some people in data science or artificial intelligence will
exploit the KTC!

~~~
VHRanger
> I asked him what he was assuming for his supply and demand curves --
> continuous, uniformly continuous, differentiable, continuously
> differentiable, infinitely differentiable, convex, pseudo-convex, quasi-
> convex, etc.? He said nothing.

> Soon I got a call from my department secretary to call my Ph.D. advisor -- I
> was out of the econ course!

What the hell? The first two semesters of microeconomic theory in graduate
school go over all of that. Supply and demand are formalized down to set
theory coming up (from Arrow's work!)

If you are interested, the Mas-Colell/Whinston/Greene text is the bible all
PhD students are forced through in microeconomics. Start at the set theory
level, define what permits construction of a utility function, and get to
defining supply and demand from there. Then get to game theory and other
topics.

You even have the theorems where free markets fail to be an efficient
mechanism of allocation! We've known those things for decades, but economics
is so politicized that it's hard for information out.

~~~
graycat
I was a grad student in applied math and had done the basic research for my
dissertation, solved the KTCQ problem, was polishing the research, and writing
the illustrative software when my advisor suggested I take an econ course.

The econ course was in the econ department, not my department, and was not an
econ grad course!

But, whatever the course was, the econ prof was apparently just terrified of
my question!

One way and another, maybe I've touched on much of what you mentioned. E.g.,
the optimization I studied, with the math rock solid, was a good start on game
theory. Later, while in a part time job, to support us while my wife finished
her Ph.D. and basically time-out from my Ph.D., I took a job in military
systems analysis with a lot in game theory. So, I dug into parts of G. Owen's
book on game theory which did the axiomatic utility function stuff and T.
Parthasarathy and T. E. S. Raghavan which did a lot of fixed point theorems,
Sion's result, Lemke's proof of Nash's result, etc. Sure, in the relatively
general game theory the job had me in, I had to consider saddlepoint results,
and apparently that is the core of equilibrium theory in econ.

Once a tried a book on math econ, and it was just a lot of elementary
regression analysis. Later I saw another such, by Tata?, and more advanced but
still regression -- a place to see more about regression than want to know,
and maybe the AI people should take a look. Later saw another such book, by
Duffie, and early on it was heavily about the Kuhn Tucker conditions. I read
the first chapter or two quickly and had some questions, went back, read
carefully, and found a counterexample for every statement in that material.

I did want to see a clean, solid, mathematical treatment of the Sharpe idea
but didn't find that -- D. Luenberger, a good mathematician, has a book on
finance that may have such a treatment.

Thanks for the reference on micro. I copied that in my place for such things
and will look at it if I get interested in econ after I exit from my startup!

Well, _nil nisi bonum._

~~~
VHRanger
Most of the advanced math in economics in current research is either in
econometrics or game theory. Not saying econ is math-less, far from it, but
most of the time we can't simply solve our problems by applying advanced math
like physics can.

Equilibrium concepts in game theory are a tough thing. The holy grail is still
getting a unifying concept of a "stable equilibrium" (see Kohlber & Mertens
'86) which is pretty much a guaranteed Nobel (but I'm not sure it even exists,
so many geniuses worked on the problem without success).

