Talk:Fundamental theorems of welfare economics

Introduction summary

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)

WTF??? Local Nonsatiation is a weak requirement???

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)

As a matter of fact, local non-satiation is even weaker than that. Local non-satiation states that for every bundle, there exists another bundle that is preferred to that bundle, in epsilon-distance to the original bundle. That means if one "good" is a "bad" in some area, the existence of a close bundle with less of the "bad" still satisfies local non-satiation. The two most prominent failures of local non-satiation (to illustrate), is when goods are only available in discrete chunks, or if there exists a global satiation point (think of CB preferences).

As a side-point, the IP user fails to explain his examples a bit more. Both toll road & internet cable monopoly offerings don't satisfy some of the assumptions of the 1st FTWE (no externality, no price-taking). Don't know why they think they can contribute if they haven't read even the first chapter in a microeconomics textbook... Oragonof (talk) 08:55, 23 June 2023 (UTC)

Hoppe of the Austrian school has proven the second theorem incorrect.

"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)

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)

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)

Deleted spurious conjecture in introduction

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)

Really Bad

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

Major Problems with assumptions!

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)

cleanup

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)

of, not, in

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)

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)

Requested move

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)

Done. —Nightstallion (?) ^{Seen this already?} 09:03, 21 March 2006 (UTC)

Change the preference relations to "succ"

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)

Not the most general or concise formulation

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)

No precise statement of the model

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)

These sound like excellent suggestions. I will take a look when I get a moment. Jonpatterns (talk) 14:49, 12 February 2018 (UTC)

Arrow's Theorem

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)

History?

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.

History

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)

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)

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)

Volunteer Marek’s edits (Nov 2020)

@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 wrote that ‘There are no externalities [footnote: ie. firms can’t pass their costs onto other people without their consent] and no transaction costs’. This has been reduced to ‘There are no externalities’. This modification is justified in the edit summary by the words ‘incorrect OR’. The new wording is incorrect on two counts. Firstly, the absence of transaction costs is indeed a requirement for the FFT. Secondly the absence of externalities is not, without qualification, a requirement at all. The reason is that the term ‘externality’ does not have a perfectly clear meaning, but is often understood as including ‘pecuniary externalities’ (see Mas-Colell et al p352: they explicitly point out that an externality of this sort ‘creates no inefficiency’). It is only the non-pecuniary externalities which interfere with the optimality of markets. Rather than get bogged down in a discussion of the terminolgy of externalities I simply explained the meaning of the term in a footnote in a way which corresponds to the assumption needed by the FFT. Without this qualification the statement of the FFT is incorrect.

• I wrote that ‘competitive markets ensure an efficient allocation of resources in the short run’, citing an observation of Hotelling’s which shows that they need not be efficient in the long run. Volunteer Marek deleted the words ‘in the short run’ (‘more OR’). I have read scores of things about the FFT none of which looks beyond short-run effects: the FFT refers to exchange and (to some extent) production, but not to the life history of the firm. I am not aware of any serious argument that competition is necessarily optimal in the long run. I cited Hotelling in support of my wording. How is that OR?

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)

Pretty much everything I removed was either original research, editorializing or simply wrong.

1) With regards to externalities - I removed it because there are both negative and POSITIVE externalities and both violate the assumption of the theorem. To equate "externalities" with "firms can't pass their costs onto other people" is to (very clumsily) characterize all externalities as negative externalities. Furthermore, what does the sentence "firms can't pass their costs onto other people without their consent" even mean? What is "consent" here?

2) Absence of transaction costs is a requirement for the FFT? Maybe in some specific way but if so then that way is usually covered by the "no externalities" assumption. Anyway. Show me a source.

3) "Pecuniary externalities" aren't really externalities, that's all there is to it (though indeed, the wording "pecuniary externalities" is confusing which is why it's mostly avoided by economists)

4) Your comment on Hotelling is clearly WP:OR and WP:SYNTHESIS. First, Hotelling here would be a primary source (he's not the last word on this). Second, I'm not sure you're even understanding Hotelling correctly. Third, what does "short run" even mean here? The distinction between short run vs. long run has several definitions (whether prices are flexible, whether capital can be accumulated, whether entry into markets is allowed), so as is this statement makes no sense. Anyway, given its assumptions, FFT apply to any economy with finite number of trading periods, or to an economy with infinite number of trading periods but with complete markets.

Volunteer Marek 16:55, 10 November 2020 (UTC)

1) You’re splitting hairs. Costs can be negative if you need to be finickety. ‘Passing on costs without people’s consent’ is distinguished from ‘passing on costs as part of a transaction accepted voluntarily by both parties’. I think any reader will undrestand this.

2) Ref [1] in the article (p7).

3) Pecuniary externalities are externalities according to Mas-Colell et al, whose book the article is based on. Also according to the wikipedia definition. I don’t see how a meaning of externality which excludes them can be taken for granted.

4) Can you show me a proof (or serious argument) that the FFT as normally understood covers the process by which firms and industries come into being and close down? If not, a statement of the FFT which does not caveat itself accordingly goes beyond anything the proofs establish. I have read Arrow, Debreu, Mas Colell etc. and all I see is exchange and (to some extent) production. Colin.champion (talk) 17:31, 10 November 2020 (UTC)

1) I'm not. Anyway. Show me a source to support that text.

2) That does not support the text you're trying to add to the article (the issue is about constrained Pareto efficiency)

3) Show me where this is related to the FFT.

4) I'm not going to engage in original research here and neither should you. Volunteer Marek 02:44, 11 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)

Notation in the proofs.

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)

Microeconomics 2

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