7 ms·

Kenneth Arrow and the golden age of economic theory

73 points by Hooke 10y ago

georgeglue1 10y ago
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
deleted 10y ago
[deleted]
graycat 10y ago
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 10y ago
  > 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 10y ago
    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.
paulpauper 10y ago
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.
- VHRanger 10y ago
  I agree that random walks are the most sensible way to model the financial markets, IMO. Anyone telling you otherwise you should be suspicious of (are they can beat the market?) Especially now with the mountain of evidence that pretty much no one can consistently beat a good market index.
  - lost953 10y ago
    The only issue with that is there are a few groups that have consistently beat the market over time, so perhaps its more pseudorandom than we think. One example might be the Renaissance Medallion Fund.
    - VHRanger 10y ago
      Right, and I generally respect Renaissance. I think people who actually can beat the market fall into the general category of "knows something others don't". Renaissance probably is one of the very few firms that could qualify as that on purely technocratic means (instead of the usual, which is shades of grey in what's really insider trading). The other category would be simply HFT, but I'm not sure how profitable that is at this point, though. Most of the people who seem to beat the market, though, are generally just happy benefactors of survival bias (eg. the 10000 monkeys on typewrites effect)
unknown_apostle 10y ago
« 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.

73 points by Hooke 10y ago

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

deleted 10y ago

[deleted]

graycat 10y ago

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 10y ago
> 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 10y ago
  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.

paulpauper 10y ago

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.

VHRanger 10y ago
I agree that random walks are the most sensible way to model the financial markets, IMO. Anyone telling you otherwise you should be suspicious of (are they can beat the market?) Especially now with the mountain of evidence that pretty much no one can consistently beat a good market index.
- lost953 10y ago
  The only issue with that is there are a few groups that have consistently beat the market over time, so perhaps its more pseudorandom than we think. One example might be the Renaissance Medallion Fund.
  - VHRanger 10y ago
    Right, and I generally respect Renaissance. I think people who actually can beat the market fall into the general category of "knows something others don't". Renaissance probably is one of the very few firms that could qualify as that on purely technocratic means (instead of the usual, which is shades of grey in what's really insider trading). The other category would be simply HFT, but I'm not sure how profitable that is at this point, though. Most of the people who seem to beat the market, though, are generally just happy benefactors of survival bias (eg. the 10000 monkeys on typewrites effect)

unknown_apostle 10y ago

« 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.