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Can we not have something short and to the point in the opening paragraph introducing these? I was thinking of something along the lines of:
I don't really understand the technical bits well enough to precis them (the above are mostly guesses), but I'm sure that somebody else does. — KayEss | talk 16:37, 12 July 2005 (UTC)
Reminds me of that old physics joke. Q: How does an elephant go down a ramp? A: Well, first assume the elephant is round... Except in economics there seems to be no compulsion to go back and adjust for missed assumptions. Let's look at a few examples. a) toll road. Is there a better alternative within epsilon? b) internet cable monopoly offering. Is there a better alternative within epsilon? c) Cheetos. Is there a better offering within epsilon? For that matter, what is the distance metric used? Is the consumption set even a metric space at all? 70.113.72.73 ( talk) 04:08, 16 July 2012 (UTC)
The above are examples of violations of local non-satiation for individual goods; however, the assumption in question is local non-satiation for the choice set as a whole. This is indeed an extremely weak assumption. All it implies is that there exists at least *one* good that the consumer could get slightly more of and would prefer slightly more of. Ossanha ( talk) 04:37, 28 April 2014 (UTC)
"Hoppe's argument dispatches entirely the notion of Pareto optimality as a social-welfare-maximizing end state. Welfare economics starts with the objective fact of self-ownership and then demonstrates that each step of voluntary acquisition and use of property satisfies the Pareto rule and thereby, improves social welfare. Moreover, each instance of state intervention into the voluntary acquisition or use of property benefits some and harms others and, thereby, fails to improve social welfare."
http://mises.org/article.aspx?Id=4014 —Preceding unsigned comment added by 70.178.234.45 ( talk) 12:15, 11 February 2010 (UTC)
Re: Hoppe ... So you are saying that everything that benefits one person and harms another reduces total utility, proof by assertion? 70.113.72.73 ( talk) 04:08, 16 July 2012 (UTC)
The claim was made that people need to understand the economy and how to make use of the lump-sum transfers for the transfers to be effective. This is false. Neither welfare theorem will hold in a hypothetical world where people don't understand how to interact in the market and buy and sell goods. —Preceding unsigned comment added by 68.42.67.48 ( talk) 06:18, 7 October 2009 (UTC)
The theorems were really hopelessly mangled. I just fixed the first theorem, though I don't have time right now to do the second (the proof is quite long). Regarding the comments below about monotonicity of preferences, that isn't actually necessary for the FFT. The weaker assumption of local nonsatiation is sufficient.
Also, why are the two theorems together in one article? I recommend splitting them into two and then just cross-referencing. Also, there seem to be a few other articles mascarading as treatments of the FFT. These should probably be deleted.
-- Ossanha
Firstly, I disagree with the use of the term "axiomatic", these propositions are based on a number of lower-level assumptions (e.g. the rationality/selfishness of all players, assuming that all traders have monotonic utility functions, and so on). The fact that this statement is proven given other assumptions shows, in my opinion, that it is not axiomatic but built on more basic axioms.
More importantly, I believe that the statement of the theorem (or perhaps of Walrasian equilibrium) is incorrect. Unless I am misinterpreting the definition, there is no assumption here that utility functions are monotonic. This assumption is necessary for a proof of the theorem.
The proof sketch shown is also flawed, due to this missing assumption. Consider the case where we have two traders. Player 1's utility functions is a constant (i.e. he/she couldn't care less about the allocation he/she receives) and Player 2's utility function is monotonic. A Walrasian Equilibrium will lie at some point on player 2's budget constraint curve. This point will not be Pareto efficient, because if Player 1 gives their goods to Player 2, Player 1 will be no worse off (their utility remains constant) and Player 2 will be better off (due to their monotonic utility function).
To summarize, this statement of the theorem is lacking the monotonic assumption, without it the theorem is false. (With it, it is provable, though with all respect I still do not like the proof sketch provided, I would like something more mathematical - personal preference)
See Feldman, "Welfare Economics and Social Choice Theory", Chapter 3 for further discussion.
I may have messed up some mathematical notation in my recent edits. The page was such a mess that whoever created it should check that. Next I'm going to change the title of the article to conform to the style manual. Michael Hardy 23:05, 6 November 2005 (UTC)
It should be Fundemental Theorems OF Welfare economics, not IN Welfare Economics. Just like Fundemental Theorem of Algebra or Calculus or whatever. Please don't change it to IN. radek 03:11, 15 March 2006 (UTC)
Support It's 'of' not 'in'. Also per comment above, good idea would be to have two seperate pages for the theorems and then this page just a short statement and redirect. radek 22:21, 19 March 2006 (UTC)
I would go through and change these myself, but I don't have the time at the moment. Every time two bundles are compared, the preference relation should be expressed rather than , and rather than . This is important because without the distinction it is hard to tell in the proofs when "greater than" is meant and when "preferred to" is meant. This makes the proofs hard to read. —Preceding unsigned comment added by 128.118.239.104 ( talk) 22:57, 14 November 2009 (UTC)
The presentation of the two theorems is not very precise mathematically, as couple comments pointed out already above. A la Debreu, the consumption sets should lie in a normed topological vector space X and price is given by an element of the dual of X. In this setting, FFT is proved in about two lines and the SFT not too much more than that. If no one objects, I might make some modifications along these lines. The short section on FFT can certainly be expanded.
Only in the general infinite-dimensional setting is the full-strength of the hyperplane-separation theorem used (although texts like Mas-Colell do freely quote it in finite-dimensional situations). The Hahn-Banach theorem, of which hyperplane separation is a corollary, holds trivially for finite dimensional topological vector spaces. Mct mht ( talk) 06:27, 27 October 2012 (UTC)
This article should at least be understandable to someone with the necessary mathematical background, not just people with knowledge of economics as well. The "formal statement" is not formal at all unless you know what "preferences", "non-satiated", "price equilibrium", "transfers", "allocation" and "Pareto optimal" mean. If it's not possible to briefly recap the mathematical model so as to make it understandable to someone with a background in pure math, then there should at least be links to articles that do contain that information. Gaiacarra ( talk) 08:48, 21 November 2017 (UTC)
Arrow's Theorem is not traditionally referred to as the "third welfare theorem," even though it is often discussed in contrast to the first welfare theorem. I can't find a single source that calls it "the third welfare theorem," and the current citation is a dead link.
The short description of Arrow's Theorem currently in the article is also problematic, it uses the unorthodox term "arrow social welfare equilibrium" without defining it, and completely misstates the independence of irrelevant alternatives assumption. (The assumption is about independence across alternatives, not independence across individuals.)
I'm going to remove the description of the "third welfare theorem," but Arrow's Theorem will still be mentioned under "related theorems." — Preceding unsigned comment added by Manybytes ( talk • contribs) 07:56, 13 August 2019 (UTC)
It's sort of strange. I've read a 700 page popular micro economics text covering this in detail, and a few other bits and pieces and it's still not clear to me who came up with these so called theorems. It's either Walras, or Arrow, or Debreu or all of them, and maybe some others too. But it is clear as mud to me exactly who contributed what when. A "History" / "Origins" section would be greatly appreciated.
I’ve added a history of the theorems, addressing the suggestion of the previous (unsigned) comment. It turned out to be longer than I expected – I always imagined that one more author would sew the topic up, but it took a long time for the ideas to evolve to their current state.
I would like to make a few changes to the wording elsewhere (though not to the maths), but I’ll wait to see if I get my head bitten off first. Colin.champion ( talk) 12:16, 26 August 2020 (UTC)
@ Volunteer Marek: Volunteer Marek has made a number of changes, chiefly deleting what he or she regards as original research on my part. It may be that some of these changes are improvements, and it may be that I’m not the best person to say that some of them introduce faults. However none of them are supported by any reasoning, and many of them seem to me to introduce significant errors. I don’t want to simply revert them, so I’ll give a couple of examples of what I consider to be errors.
I think Volunteer Marek is a little quick to dismiss as OR statements which are based on a close and careful reading of numerous authoritative sources. Colin.champion ( talk) 11:00, 10 November 2020 (UTC)
I was hoping someone else would come into the discussion. Consensus isn’t reached by two people slugging it out indefinitely. Colin.champion ( talk) 07:40, 12 November 2020 (UTC)
Maybe this notation is standard in economics (I'm sure it's standard in some field), but can we please have at least some indication for non-experts of what the mathematical notation being used means? Looking up mathematical notation conventions without knowing the names for the concepts they describe is very difficult, so could someone who knows please add at least an explanatory link for anything you wouldn't get in grade school (i.e., anything most English-speakers can't be expected to already know)? DubleH ( talk) 06:54, 31 January 2022 (UTC)
If you want to know, what I didn't understand was the meaning of the "(X*, Y*, p)" and "(X*, Y*)" tuples in the first paragraph of the first proof. I understood most of the set theory and vector stuff that followed. I havn't really looked at what's past that closely enough to know if I understand it. DubleH ( talk) 06:59, 31 January 2022 (UTC)
Fundamental theorem of welfare economics 2405:205:B086:ABD9:0:0:1967:8AC ( talk) 12:35, 6 June 2023 (UTC)
What is a selfish preference and why was it included as a hypothesis? There is no mention of this in the literature. See page 549 of the following book:
Mas-Colell, Andreu, Whinston, Michael D., and Green, Jerry R. (1995). "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680. Daniellesteffen ( talk) 21:35, 21 May 2024 (UTC)