Talk:Arrow–Debreu model

message: I don't know whether my brains is playing tricks on me, but I thought the central feature of Arrow-Debreu model is more to do with the anonymous nature of the markets. Ie Nobody care who supplies chair or steel in the market or who buy the same goods. I didn't see that from the Arrow-Debreu model article. The definition above is from "Toward new paradigm in monetary economics" by Stiglitz
I agree that the description is somewhat narrow and have added an introduction with reference to the economic theory pages on the subject. I agree with the original author(s) that in financial economics the term is typically used with reference to AD-securities.
I would like to hear more on what BIC's are (from the original author(s) or anybody else informed on that subject).--Olejasz 14:18, 9 October 2005 (UTC)
thread_title: Anonymous nature of the markets

message: It's been a long while, and there seems to be no useful addition on BICs. There's no Wikipedia page or reference within this article. So I'm pulling BICs out.Cretog8 (talk) 13:05, 25 June 2008 (UTC)
thread_title:  BICs again 

message: Please, could someone write the model? Lbertolotti  —Preceding unsigned comment added by Lbertolotti (talk • contribs) 23:56, 18 February 2009 (UTC)
thread_title:  Model 

message: An IP editor wrote the following  Kiefer.Wolfowitz 09:24, 14 April 2012 (UTC)
thread_title:  Uniqueness 

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thread_title:  External links modified 

message: I propose that Arrow–Debreu model be merged into Complete market. The model is but an expression of the complete-market notion. It all comes down to the same exact ideas, expressed differently. Wikipedia users would benefit from having the model as a (large) section of the "complete markets" article. What do others think? (Note: Discussing the merger will be done, according to the rules, in the target article's talk page, and specifically here). Take care, all. -The Gnome (talk) 09:56, 4 April 2018 (UTC)
thread_title:  Merger proposal 

message: The claim that (strict) concavity of utility functions yields uniqueness of equilibrium is ridiculous. 2603:8001:7402:7D0C:3441:449:CDB5:5402 (talk) 15:29, 24 January 2023 (UTC)
replies: text: strongly concave and twice continuously differentiable pony in a strange land (talk) 18:31, 25 January 2023 (UTC)

text: Sorry, I checked again and yes, the proper statement is that equilibrium is unique generically, but not always. It is fixed (with proper reference). pony in a strange land (talk) 18:47, 25 January 2023 (UTC)
replies: text: I looked up your reference and of course it says no such thing (not surprisingly since it is a completely false assertion). My guess is that you are referring to the well-known generic local (!) uniqueness result, emanating with Debreu (1970 as I recall). Local uniqueness means that  equilibria are isolated, so that they are finite in number, but there could be, say, a 1001 of them (it's generically an odd number).  This is sometimes called  determinacy, as opposed to there being a continuum of equilibria (indeterminacy), which is the non-generic thing (at least in finite-dimensional models). Strong convexity, a rather mild assumption, doesn't have much to do with all this. It so happens that I am an expert in the field, but any well-educated graduate student ought to know that you can't get uniqueness of equilibrium so cheaply. 2603:8001:7402:7D0C:C55B:B017:D5CD:5584 (talk) 06:34, 7 February 2023 (UTC)

text: I see that you actually wrote "strongly concave," but that is even weirder, since that would be a cardinal, not an ordinal property. It's not very difficult to draw an Edgeworth box with say three equilibria where the preferences are as nice as you might wish, as was well-known to, say, Marshall. 2603:8001:7402:7D0C:C55B:B017:D5CD:5584 (talk) 06:43, 7 February 2023 (UTC)
replies: text: A more interesting and pertinent question, which I didn't notice you addressing, is why is there a separate entry for the Arrow-Debreu model, as opposed to the general entry for general equilibrium theory. Answers to this might involve the assumption in the A-D model of a given number of producers (as well as consumers, as well as goods), as opposed to, say, the McKenzie model, with a variable number of producers. One might also contrast these finiteness assumptions against OLG models (infinite number of goods and consumers) or Aumann models (a continuum of consumers) or Mas-Colell type product differentiation models (an infinite number of goods). It turns out to make a difference, e.g. the first theorem of welfare economics can fail with an infinite number of goods or consumers. 2603:8001:7402:7D0C:C55B:B017:D5CD:5584 (talk) 07:03, 7 February 2023 (UTC)

text: It's not usually in my character to work for free, but I am worried that the younger generation has not only made few advances in economic theory in this millenium but that it has also forgotten what was formerly well known. It's not really their fault I suppose since it is now much easier to concentrate on data. I fear that all this misinformation on the internet will corrupt impressionable minds. Getting it right on this site may mean that ChatGPT  learns what is correct, which could prove important in the future (I tested and it was woefully ignorant of economic theory: I wonder where it got its ideas; perhaps it does not have access to the literature, which is often hidden behind paywalls). 2603:8001:7402:7D0C:112F:EF32:D428:33EF (talk) 18:26, 7 February 2023 (UTC)

text: I see that you are a mathematician, as well as of course a pony, though perhaps not an economist (do you then know about ordinal vs cardinal?). The generic dimension zero of the equilibrium set is a straightfoward application of the transversality theorem in differential topology (so there is some truth to the stated need for differentiablity of demand) and the finiteness part comes from a side argument where though price space is not generally compact the demands are proper mappings over that domain. This then is the generic local uniqueness of the A-D model (not necessarily true in infinite dimensional models; see my previous aside on what makes an A_D model what it is) To get true uniqueness of equilibrium you usually look at the Poincare-Hopf index theorem. which fits the situation pretty well and then make some sort of  very strong assumption forcing each zero to have index unity, in which case there can be but one (unity being the Euler characteristic of the price domain). The classical example of a strong assumption is  the assumption (Dierker) that all the equilibrium are locally stable and hence have the requisite index one, so that there can then be but one equilibrium. 2603:8001:7402:7D0C:112F:EF32:D428:33EF (talk) 18:55, 7 February 2023 (UTC)

text: Perhaps you are getting strong concavity from consideration of what it takes to get a unique maximum to the optimization of a real-valued function, but here we are dealing with an equilibrium problem in multiple dimensions, not a maximization one (though rationality/maximization for the economic agent (consumer, producer) and then equilibrium for the entire economy are indeed the overriding principles of neoclassical economics). 2603:8001:7402:7D0C:112F:EF32:D428:33EF (talk) 19:13, 7 February 2023 (UTC)

text: Bothering to scroll up I see that the reference to uniqueness was already called into question in 2012! Of course, I am not calling into question such claims, I am telling you they are utterly wrong (that is no minor adjustment is going to fix up what is so very far from the truth). Luckily, I have the impression that no one has paid any attention to this site in a long time, else the "wisdom of the masses" would have prevailed long ago. 2603:8001:7402:7D0C:112F:EF32:D428:33EF (talk) 20:11, 7 February 2023 (UTC)

text: With a little effort one can locate many other problems with this exposition. While I agree that convexity is typically used in employing the Arrow-Debreu model, I wouldn't say it was necessary and certainly the mathematical statement that fixed point theorems fail without convexity is untrue. Let me point out the Eilenberg-Montgomery fixed point theorem which uses contractibility, a weaker notion than convexity and was employed by Debreu in his 1952 PNAS article. There are plenty of other dubious things in the article, though perhaps nothing as glaring as the statement about uniqueness. 2603:8001:7402:7D0C:112F:EF32:D428:33EF (talk) 03:24, 8 February 2023 (UTC)
thread_title:  uniqueness