John Quiggin gives a modest defence of existence theorems in economics, one of the three real vices of economists according to Deirdre McCloskey.
Existence theorems, for McCloskey are the archetypal example of ‘blackboard economics’, mathematical games yielding purely qualitative results that can be overturned with modest changes in assumptions. They were the high point of mathematical economics in the 50s and 60s … There are a wide variety of ‘impossibility theorems’ demonstrating the non-existence of index numbers with various properties [an area of research interest for John]. Familiarity with such theorems can save a lot of pointless effort, and they are therefore worth looking for. But an impossibility theorem is just the negative form of an existence theorem (or, if you prefer, an existence theorem proves the impossibility of the corresponding impossibility theorem).
This is a rather prosaic defence, that certainly does not justify the high status accorded to the kind of theory exemplified by existence theorems. But the argument can be pushed a bit further by considering the most famous impossibility theorem, that of Arrow who showed (roughly speaking) that no voting system having a set of seemingly desirable properties could work for all possible sets of voter preferences. This impossibility theorem precluded a lot of potential effort in designing ideal voting systems. [Emphasis added.]
This is a nice parallel. Actually, it’s so nice that it may prove more than John intended. (I absolve him of responsibility for what follows.)
Arrow’s Impossibility Theorem illustrates the point neatly. We begin with the assumptions. Roughly speaking, Arrow made a list of four criteria that reasonable people might think any method of aggregating people’s preferences ought to have. These conditions are (1) That the full range of everyone’s preferences be considered, (2) That if everyone prefers x over y then the group decision should as well, (3) That the position of x relative to y in the group prefernece depends only on the position of x relative to y in each individual’s preference, and (4) That there isn’t a dictator — i.e., someone who gets to have their preference enforced over everyone else’s. We then move forward to the impossibility result: there is in fact no method satisfying all four criteria. Any voting system would necessarily violate at least one of them. The proof is striking because the initial assumptions are so plausible, even weak, but they cannot all be satisfied together and so we find that the desirable result is impossible. And so we give up our quest for what we now realise is a chimera —- the idea of a perfect method of aggregating individual preferences.
As John says, existence theorems are the negative form of impossibility theorems. The classic existence theorems in economics — such as those for general equilibrium, also due to Kenneth Arrow, along with Gerard Debreu — illustrate the point neatly. We begin with the result. Roughly speaking, Arrow and Debreu wanted to show that supply and demand could be in balance in all markets at once. We then move backward to the assumptions necessary to make possible such a result. These include (1) All individuals are perfectly rational, (2) All trades take place simultaneously and instantaneously, (3) There is perfect information about all markets for all products in all conditions both now and at any point in the future, (4) Money does not exist. With these (and other) assumptions in place, the existence of a general equilibrium can be proved. The proof is striking because the initial assumptions are so implausible, even absurd, but they must all be satisfied together in order for the desirable result to be possible. And so we give up our quest for what we now recognise is a chimera — the idea that our world could ever contain economies capable of general equilibrium.
Whoops. I suppose many economists wouldn’t take that last step along the road. For their own reasons, most economists do not treat existence theorems of this sort in the obvious way — i.e., as a kind of useful reductio ad absurdum, or at least ad ridiculum. I sometimes wonder why economists so rarely adopt this interpretation. (I hear the phrase “F-twist” on the breeze.) The most plausible interpretation of the results is, “Well, we got the desired result … But look at the assumptions we had to make to do it. Absurd. The result can therefore never obtain. QED.”
Update: John Quiggin responds. Maybe I was wrong about the realism of the Arrow-Debreu temporality assumptions, because his comments seem to have appeared pretty much instantaneously.
Just a small point: that a result follows from certain assumptions doesn’t imply that those assumptions are necessary to generate the result. That may, of course, be true, but it’s being true (if it is) is not part of the proof of the result in question. In general, reductios work when the absurdity is a necessary condition of the result to be reduced, not when it is a sufficient condition. I might just be missing something.
that a result follows from certain assumptions doesn’t imply that those assumptions are necessary to generate the result.
Yes, that’s right, I fudged over that in the post. (I plead “It’s a blog, not the AER.”) The promise of many such theorems is that we’ll find a way to prove a more general case later. Often this never happens.
But now (after reply to Juan) your terminology (“existence theorems”) is confusing.
Amplifying on Juan’s point — the result that IF such and such, then there is a general equilibrum, is an “existence theorem”. The result that the existence of a general equilibrium REQUIRES such and such is NOT an “existence theorem.” It’s really of the form “if there is a general equilibrium, then such and such” and that assumes existence as the antecedent of the if-then, rather than proving it as the consequent. But this means it’s really already an impossibility theorem in disguise.
For, notice that:
Conditions C are necessary for a K to exist
is equivalent to
If a K exists, then conditions C hold
which is equivalent to
If conditions C fail to hold, then no K exists
which is an IMPOSSIBILITY theorem.
I think that what you’re really noticing is that the search for an existence theorem can lead to both necessary AND sufficient conditions for the existence of something, in other words, a claim of the form:
a K exists if and only if conditions C obtain
which is really both an existence theorem (if conditions C, then a K exists) and an impossibility theorem (if conditions C fail, then no K exists) at once.
This, however, isn’t a defence of existence theorems, but of the search for them.
In mathematics I have read people talking about certain solutions or proofs being “possible in principle”, in the sense that they are not impossible like squaring the circle, but would merely require a computer half the size of the known universe to solve. Mathemeticians also speak of “very large numbers” which are still not infinite and thus can be handled just the same as ordinary numbers: for example, if you take the estimated number of electrons in the universe, square it, and keep squaring the result 100 times or so.
I agree with the skeptical conclusion that proving that something is formally possible isn’t proving much in a science which is meant to have any real- world (world of historical time, world of historical experience) applications.
I disagree with most of the comments here. The existence of general equilibria is not a priori obvious under any conditions. If general equilibria didn’t exist even for perfect markets, then that would be a significant fact.
I didn’t know about the “no money” condition, though. How does the proof depend on that?
À Gauche
Jeremy Alder
Amaravati
Anggarrgoon
Audhumlan Conspiracy
H.E. Baber
Philip Blosser
Paul Broderick
Matt Brown
Diana Buccafurni
Brandon Butler
Keith Burgess-Jackson
Certain Doubts
David Chalmers
Noam Chomsky
The Conservative Philosopher
Desert Landscapes
Denis Dutton
David Efird
Karl Elliott
David Estlund
Experimental Philosophy
Fake Barn County
Kai von Fintel
Russell Arben Fox
Garden of Forking Paths
Roger Gathman
Michael Green
Scott Hagaman
Helen Habermann
David Hildebrand
John Holbo
Christopher Grau
Jonathan Ichikawa
Tom Irish
Michelle Jenkins
Adam Kotsko
Barry Lam
Language Hat
Language Log
Christian Lee
Brian Leiter
Stephen Lenhart
Clayton Littlejohn
Roderick T. Long
Joshua Macy
Mad Grad
Jonathan Martin
Matthew McGrattan
Marc Moffett
Geoffrey Nunberg
Orange Philosophy
Philosophy Carnival
Philosophy, et cetera
Philosophy of Art
Douglas Portmore
Philosophy from the 617 (moribund)
Jeremy Pierce
Punishment Theory
Geoff Pynn
Timothy Quigley (moribund?)
Conor Roddy
Sappho's Breathing
Anders Schoubye
Wolfgang Schwartz
Scribo
Michael Sevel
Tom Stoneham (moribund)
Adam Swenson
Peter Suber
Eddie Thomas
Joe Ulatowski
Bruce Umbaugh
What is the name ...
Matt Weiner
Will Wilkinson
Jessica Wilson
Young Hegelian
Richard Zach
Psychology
Donyell Coleman
Deborah Frisch
Milt Rosenberg
Tom Stafford
Law
Ann Althouse
Stephen Bainbridge
Jack Balkin
Douglass A. Berman
Francesca Bignami
BlunkettWatch
Jack Bogdanski
Paul L. Caron
Conglomerate
Jeff Cooper
Disability Law
Displacement of Concepts
Wayne Eastman
Eric Fink
Victor Fleischer (on hiatus)
Peter Friedman
Michael Froomkin
Bernard Hibbitts
Walter Hutchens
InstaPundit
Andis Kaulins
Lawmeme
Edward Lee
Karl-Friedrich Lenz
Larry Lessig
Mirror of Justice
Eric Muller
Nathan Oman
Opinio Juris
John Palfrey
Ken Parish
Punishment Theory
Larry Ribstein
The Right Coast
D. Gordon Smith
Lawrence Solum
Peter Tillers
Transatlantic Assembly
Lawrence Velvel
David Wagner
Kim Weatherall
Yale Constitution Society
Tun Yin
History
Blogenspiel
Timothy Burke
Rebunk
Naomi Chana
Chapati Mystery
Cliopatria
Juan Cole
Cranky Professor
Greg Daly
James Davila
Sherman Dorn
Michael Drout
Frog in a Well
Frogs and Ravens
Early Modern Notes
Evan Garcia
George Mason History bloggers
Ghost in the Machine
Rebecca Goetz
Invisible Adjunct (inactive)
Jason Kuznicki
Konrad Mitchell Lawson
Danny Loss
Liberty and Power
Danny Loss
Ether MacAllum Stewart
Pam Mack
Heather Mathews
James Meadway
Medieval Studies
H.D. Miller
Caleb McDaniel
Marc Mulholland
Received Ideas
Renaissance Weblog
Nathaniel Robinson
Jacob Remes (moribund?)
Christopher Sheil
Red Ted
Time Travelling Is Easy
Brian Ulrich
Shana Worthen
Computers/media/communication
Lauren Andreacchi (moribund)
Eric Behrens
Joseph Bosco
Danah Boyd
David Brake
Collin Brooke
Maximilian Dornseif (moribund)
Jeff Erickson
Ed Felten
Lance Fortnow
Louise Ferguson
Anne Galloway
Jason Gallo
Josh Greenberg
Alex Halavais
Sariel Har-Peled
Tracy Kennedy
Tim Lambert
Liz Lawley
Michael O'Foghlu
Jose Luis Orihuela (moribund)
Alex Pang
Sebastian Paquet
Fernando Pereira
Pink Bunny of Battle
Ranting Professors
Jay Rosen
Ken Rufo
Douglas Rushkoff
Vika Safrin
Rob Schaap (Blogorrhoea)
Frank Schaap
Robert A. Stewart
Suresh Venkatasubramanian
Ray Trygstad
Jill Walker
Phil Windley
Siva Vaidahyanathan
Anthropology
Kerim Friedman
Alex Golub
Martijn de Koning
Nicholas Packwood
Geography
Stentor Danielson
Benjamin Heumann
Scott Whitlock
Education
Edward Bilodeau
Jenny D.
Richard Kahn
Progressive Teachers
Kelvin Thompson (defunct?)
Mark Byron
Business administration
Michael Watkins (moribund)
Literature, language, culture
Mike Arnzen
Brandon Barr
Michael Berube
The Blogora
Colin Brayton
John Bruce
Miriam Burstein
Chris Cagle
Jean Chu
Hans Coppens
Tyler Curtain
Cultural Revolution
Terry Dean
Joseph Duemer
Flaschenpost
Kathleen Fitzpatrick
Jonathan Goodwin
Rachael Groner
Alison Hale
Household Opera
Dennis Jerz
Jason Jones
Miriam Jones
Matthew Kirschenbaum
Steven Krause
Lilliputian Lilith
Catherine Liu
John Lovas
Gerald Lucas
Making Contact
Barry Mauer
Erin O'Connor
Print Culture
Clancy Ratcliff
Matthias Rip
A.G. Rud
Amardeep Singh
Steve Shaviro
Thanks ... Zombie
Vera Tobin
Chuck Tryon
University Diaries
Classics
Michael Hendry
David Meadows
Religion
AKM Adam
Ryan Overbey
Telford Work (moribund)
Library Science
Norma Bruce
Music
Kyle Gann
ionarts
Tim Rutherford-Johnson
Greg Sandow
Scott Spiegelberg
Biology/Medicine
Pradeep Atluri
Bloviator
Anthony Cox
Susan Ferrari (moribund)
Amy Greenwood
La Di Da
John M. Lynch
Charles Murtaugh (moribund)
Paul Z. Myers
Respectful of Otters
Josh Rosenau
Universal Acid
Amity Wilczek (moribund)
Theodore Wong (moribund)
Physics/Applied Physics
Trish Amuntrud
Sean Carroll
Jacques Distler
Stephen Hsu
Irascible Professor
Andrew Jaffe
Michael Nielsen
Chad Orzel
String Coffee Table
Math/Statistics
Dead Parrots
Andrew Gelman
Christopher Genovese
Moment, Linger on
Jason Rosenhouse
Vlorbik
Peter Woit
Complex Systems
Petter Holme
Luis Rocha
Cosma Shalizi
Bill Tozier
Chemistry
"Keneth Miles"
Engineering
Zack Amjal
Chris Hall
University Administration
Frank Admissions (moribund?)
Architecture/Urban development
City Comforts (urban planning)
Unfolio
Panchromatica
Earth Sciences
Our Take
Who Knows?
Bitch Ph.D.
Just Tenured
Playing School
Professor Goose
This Academic Life
Other sources of information
Arts and Letters Daily
Boston Review
Imprints
Political Theory Daily Review
Science and Technology Daily Review