Talk:Fundamental theorems of welfare economics

message: 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:
:The Two Fundamental Theorems of Welfare Economics are two well-known axiomatic propositions in welfare economics. The first proves the efficiency of free markets and the second that in a free market it is possible to redistribute wealth.
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)
thread_title: Introduction summary

message: "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)
replies: text: The theorem is mathematically correct given the assumptions. Go ahead and prove that the argument is invalid given the assumptions and you'll be in line for the Nobel Prize. Whether the assumptions are realistic is another question. I don't think any modern academic economist thinks they always hold. Whether they are a good approximation depends on the case in question... Oragonof (talk) 09:01, 23 June 2023 (UTC)
thread_title:  Hoppe of the Austrian school has proven the second theorem incorrect. 

message: 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)
thread_title:  Deleted spurious conjecture in introduction 

message: 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.
:You don't need monotonicity for the first theorem. All you need is local non-satiation. See Mas-Colell, Whinston and Green chapter 16. radek 19:02, 17 January 2006 (UTC)
thread_title:  Major Problems with assumptions! 

message: 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)
thread_title:  cleanup 

message: 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)
replies: text: Agreed, I tried moving too "of", but I got an error message, hopefully someone can help fix this.--Bkwillwm 19:37, 16 March 2006 (UTC)
thread_title:  of, not, in 

message: 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)
replies: text: Done. —Nightstallion (?) Seen this already? 09:03, 21 March 2006 (UTC)
thread_title: Requested move

message: 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 x_i \succ y_i
rather than x_i > y_i, and x_i \succeq y_i rather than x_i \geq y_i.  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)
thread_title: Change the preference relations to "succ"

message: 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)
thread_title:  Not the most general or concise formulation 

message: 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)
replies: text: These sound like excellent suggestions. I will take a look when I get a moment. Jonpatterns (talk) 14:49, 12 February 2018 (UTC)
thread_title:  No precise statement of the model 

message: 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)
thread_title:  Arrow's Theorem 

message: 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)
replies: text: My head still being in place, I rewrote the final section (‘Related theorems’). I removed the claim that Arrow’s ‘Impossibility theorem’ is a third fundamental theorem on the grounds that no one had spoken up for it in the course of a year, and provided a slightly longer summary of the Greenwald-Stiglitz theorem in compensation. Whether the G-S theorem merits its place other people can decide. Colin.champion (talk) 13:47, 28 August 2020 (UTC)
replies: text:  I reworded the lead para and the section on ‘interpretations’, merging the latter into the former and generally being more guarded in tone. I think the article was originally written in the belief that ‘lump sum payment’ meant ‘one-off payment’, but while Mas-Colell et al. do not say what it means, it is clear that the ‘lump-sum transfers’ on p331 of their book must be recurring per capita payments. Colin.champion (talk) 14:36, 29 August 2020 (UTC)
thread_title:  History 

message: 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)
thread_title:  Notation in the proofs. 

message: Fundamental theorem of welfare economics 2405:205:B086:ABD9:0:0:1967:8AC (talk) 12:35, 6 June 2023 (UTC)
thread_title:  Microeconomics 2 

message: 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)
replies: text: Correct. No such mention in the literature. Must be removed. Limit-theorem (talk) 11:20, 24 May 2024 (UTC)
thread_title:  First Fundamental Theorem, 5th bullet ("Preferences are selfish.")