A correspondent asks, of my claim that the market's social welfare function weights everybody by the inverse of their marginal utility of wealth--and hence (if you are willing to grant that your marginal utility of wealth is lot less than that of a Bengali peasant) you receive a much higher weight in the market system's implicit preferences than the guy behind the water buffalo in the Ganges delta:

Brad: I assume this is a standard theorem. Do you know of an on-line reference?

Actually, I cannot recall seeing it anywhere, although it is a completely trivial result. I remember thinking of it while taking William Thomson's very nicely-taught course... what was it called?... at Harvard in... the spring of 1981, I think it was?

So, anyway, here is an online reference:

Posted by DeLong at October 9, 2003 08:33 PM | TrackBack

Comments

How do you know the partial derivatives exist at all? Continouity of the utility functions, which I guess you have assumed implicitely, doesnīt guarantee differentiability.

The assumption that a allocation is internal is also pretty strong. I doubt there are many individuals consuming positive amounts of everything one can buy in the whole wide world.

Thanks! A nice presentation. I can't imagine that, if I ran a blog, I'd be willing to write a three page explanation for the market SWF just for a bunch of groupies who had nothing better to do than hang around my webpage sniping at each other, and I admire your resolve.

(No offense intended to either myself or my fellow commentators here.)

Posted by: Julian Elson on October 9, 2003 09:38 PMWell, from my point of view there's little resolve to admire: I'll use it someday in something.

And then there is the *real* reason for the weblog: to get the "groupies" busy indexing it, so that I can find things later via google...

:-)

Posted by: Brad DeLong on October 9, 2003 09:57 PMMichael Greinecker beat me to the punch; we don't even know that the U[j] are continuous, let alone differentiable. In fact, they aren't continuous, since there are some levels of "consumption" which lead to death--in your symbols there are values of the M[i](j) vector (that is, the whole list of things consumed by a person) for which U[j](M[i](j)) will abruptly drop to zero--the person starves or dies for lack of air or what have you.

Can some argument be made that the U(...) are at least integrable be made? And, given that, can the proof be recast in terms of integrals? Or, conversely can it be shown that there are conditions required for a utility metric to be well defined?

This problem looks not-so-trivial after all, and not-so-trivial in an interesting way.

Posted by: Randolph Fritz on October 9, 2003 10:04 PMThe problem of death isnīt that big. One can incorporate that in the consumption set. It makes things more complicated though.

The conditions on preferences for the existence of continous utility functions are rather harmless IMO.

Iīm not sure about production economies, but it is relatively simple to show in a pure exchange economy that the set of (individually rational) allocations is compact, so a maximum=pareto optimum usually exists. The details can be found in Aliprantis,Burkinshaw,Brown 1989.

Whithout differentiability, the concept of marginal utility becomes of course meaningless, so the theorem isnīt IMO that useful for welfare economics.

My main problem with this theorem is that it gives a false impression of what prices do. Itīs like the difference between the MRS=MRS proofs of the second theorem of welfare economics and the one given by Arrow.

Btw: While my remarks are rather critical, I appreciate the effort of Brad. It is great to have a place to discuss such things.

Posted by: Michael Greinecker on October 9, 2003 10:20 PMYes. The person does starve to death. This is, in a way, Amartya Sen's point about famines--that if you have nothing valuable to sell (i.e., landless laborers in Bengal when World War II has disrupted international trade)--the value of your wealth is zero, your marginal utility of wealth is infinite, and so you have an infinitesimally small weight in the market's SWF, and you starve to death.

This is how the market works.

Posted by: Brad DeLong on October 9, 2003 10:22 PMElson>>groupies ... hang[ing] around my webpage sniping at each other>> :-D

DeLong>>I cannot recall seeing it anywhere, although it is a completely trivial result.

Trivial indeed, but IMO equally important. If we call the system optimal but still see starving poor within it - that's trivially the same as saying the system don't care about the poor. You don't really need math for that, but saying it in mathematical language underlines it in a wonderfully provocative fashion. Brad DeLong has to be admired for this, its creative yet careful, and very stylish. Its one really good reason to be a hangabout on this site.

Now the sniping:

Greinecker>>Continouity ... doesn't guarantee differentiability. ... doubt there are many individuals consuming ...everything ... in the world.>> Well, beeing this rigorous, you could easily wreck all the neoclassic economics.

Fritz>>[Utilies of wealth] aren't continuous, since there are some levels of "consumption" which lead to death>> As DeLong implies in his comment above, it suffices to measure utility above the deadly starvation limit, where it may well be approximately (not less precise than most macro stuff) differentiable.

Posted by: Mats on October 10, 2003 01:13 AMI was going to remark on the earlier thread that when a labor unit starves to death, from an economist's point of view that's the same as a firm going bankrupt or a mine going out of production, but there are other points of view from which it is not the same. While I rather grudgingly admire the power, accuracy, and usefulness of economic analysis, I tend to be wary of its blind spots. The most infuriating rightwingers I meet are successful or aspiring entrpreneurs who've taken 2 or 3 economics courses and believe in them the way Falwell believes in scripture. For them there is no problem with Gates' utility function being eight million times that of a fruitpicker supporting a family, since Gates is eight million times as productive. In fact, to them eight million probably seems small.

Posted by: Zizka on October 10, 2003 03:46 AMWell... the assumptions are admittedly strong, but then again, it's a really easy, short proof as well.

This somehow makes me think that you could do a bigger, harder proof with weaker assumptions that would completely fly over my head, though I actually have no rigorous, logical reason for believing that to be the case.

Posted by: Julian Elson on October 10, 2003 07:28 AM"Well, beeing this rigorous, you could easily wreck all the neoclassic economics."

The neoclassical theory holds up to the highest standards of mathematical rigor.

"This somehow makes me think that you could do a bigger, harder proof with weaker assumptions that would completely fly over my head, though I actually have no rigorous, logical reason for believing that to be the case."

Again: Without differentiability one can say nothing about marginal utility. There is no general version of this.

Posted by: Michael Greinecker on October 10, 2003 07:38 AM>>The neoclassical theory holds up to the highest standards of mathematical rigor.>>

yes it does, so let me rephrase:

Greinecker>>Continouity ... doesn't guarantee differentiability. ... doubt there are many individuals consuming ...everything ... in the world.>> Well, beeing this keen on avoiding strong claims, you could easily wreck all the neoclassic economics.

You doubt that utility is differentiable, while accepting that there are no benefits of scale (real world economy is built on benefits of scale). You accept that all goods are infinitely divisable, a car, a washing machine? You accept that preferences are constant (large part of the advertising industry is made to reverse preferences)?

Where did I state that I think the Walrasian model to be realistic? I see it, in the fashion of Frank Hahn, as a benchmark that helps us to analyse the inefficiencies of the real world.

One can approximate many things with smooth functions, differentiability for interior solutions isnīt the biggest problem. But having no corner solutions renders the analysis pretty much completely useless. One gets a wrong intuition from this.

In his groundbreaking paper on the two welfare theorems, Arrow argued that his generalization to corner solutions shows "[...] the role of prices in allocation is more fundamental than the equality of marginal rates of substitution or transformation, to which it is usually subordinated."

Btw: Infinite divisibility isnīt much of a problem in large economies.

Was the page deleted? I got a blank screen.

Posted by: Dick Thompson on October 10, 2003 09:22 AMZizka may have implicitly beaten me to the punch, but just to make sure I understand correctly, wealth is exogenous in this model, right?

Posted by: Tom on October 10, 2003 09:41 AMBrad, all your sixes are backwards.

For those that couldn't read it by clicking, like me, right click and save target. Then just load it in your Acrobat reader from your hard drive.

Posted by: Max Sawicky on October 10, 2003 09:45 AMGreinecker

First you imply that its too strong to claim that utility should be differentiable: "Whithout differentiability, the concept of marginal utility becomes of course meaningless, so the theorem isnīt IMO that useful"

Then you seem to accept that all that we have are essentially based on rather strong claims: "did I state that I think the Walrasian model to be realistic?".

Now you think the analysis is "pretty much completely useless" because we have "no corner solutions". I'll give you a corner solution that many conservatives would love - Bill Gates takes it all and the rest of us starve to death.

Posted by: Mats on October 10, 2003 09:50 AMHey, Brad, thanks for the paper. I'm impressed that you put that together. Note that it's okay with me if you actually refer to me by name, not just as "a correspondant", or for that matter if you linked to my website, thus accidently sending hits my way.

Did you use TeX for the paper? I'm sitting in on a graduate econ class here at the University of Washington, and the econ students here have never heard of TeX. Many of them type up their homework using Word. That way lies madness, I tell you, madness!

Posted by: Walt Pohl on October 10, 2003 10:12 AMYou can have differentiability only on the interior, so yo would have to assume that the underlying set is open, which is hard to justify and allows for equilibria not to exist.

You can force interior solutions to exist by conditions on the preferences. So the less you have of a good, the more it must mean to you. But that is pretty ad hoc and seems to be absurd when we have specified commodities halfway precise. If substitutes exist, it seems to be very unreasonable. You wonīt spend all your savings on the first drop of Coca Cola when you can buy Pepsi for the regular price.

BTW: If that is somehow your impression, Iīm surely not a conservative.

Brad --

Thanks for a thought-provoking article. One thought it provoked in my case is a question: How might one compare the social welfare function of the market with that of alternative processes of social choice? On the face of it, it seems that the market's welfare function sucks, but the welfare function of the political process might suck even more.

Consider that form the political process's point of view, your weight is proportional to your marginal probability of changing the outcome of important elections, by the relevent measure of 'important'.

This means that if you don't live in a swing state or in a swing district, your weight as an individuum is zero. If you're too conservative or too liberal to swing between candidates, ditto. If you're living in a small country, your weight may be relatively high in the internal politics of that small country, but close to zero as far as international politics are concerned. The list goes on, and I suspect the resulting welfare function wouldn't look pretty. But so far I don't know how to test that.

With this in mind, have you tried to calculate the inequalities generated by the political process? If so, how does it compare to the market? If not, do you know of anybody else who tried it? I'm curious!

Posted by: Thomas Blankenhorn on October 10, 2003 10:34 AM"One thought it provoked in my case is a question: How might one compare the social welfare function of the market with that of alternative processes of social choice? On the face of it, it seems that the market's welfare function sucks, but the welfare function of the political process might suck even more."

Well, under the conditions usually assumed, the follwing holds: Every Pareto optimal allocation can be the market equilibrium of some initial distribution. That means basiically all efficient situations are like market situations.

Modeling such a political process as described by you is anything but easy.

Posted by: Michael Greinecker on October 10, 2003 10:44 AMWhy a linear SWF? Actually, one problem with this whole exercise is that the utility functions are most likely ordinal and not cardinal. One minor issue and one tougher issue. Suppose we had two consumers - me and Bill Gates. Some apologist for uneven income distribution might argue that Bill's marginal utility (first derivative) from consuming the first dollar is much higher than mine. But then maybe my SWF weight (the lambda) is higher than his. That's the minor concern I have. The case tht uneven income distribution is bad given this linear SWF hinges on declining marginal utility (negative 2nd derivative). Suppose again - an apologist for uneven income distribution suggests the 2nd derivative is not negative. A Benthamite SWF might be swayed but John Rawls would not be. And somewhere between a Benthamite SWF and a Rawlsian SWF (both consistent with Pareto's thinking) lies a whole host of nonlinear SWFs that would say unequal income distribution is undesirable even if the utility function does not exhibit a falling first derivative. At least, I think this is what Fred Westfield told me 25 years ago when I was musing over similar ideas.

Posted by: Hal McClure on October 10, 2003 11:07 AM> Well, under the conditions usually assumed, the follwing holds: Every Pareto

> optimal allocation can be the market equilibrium of some initial distribution.

> That means basiically all efficient situations are like market situations.

That's interesting -- thanks. But I'm not sure how relevant it is to Brad's problem because we can't choose the initial distribution. So it's logically possible that the market equilibrium emerging from the initial distribution we have is not Pareto-efficient. Moreover, the equilibrium might also be undesirable by standards other than Pareto efficiency. If you starve and I'm rich, that can still be Pareto-efficient because your not starving requires me being made worse off.

I think Hal McClure gets it right when he says:

"Actually, one problem with this whole exercise is that the utility functions are most likely ordinal and not cardinal."

This means the whole marginal utility of wealth thing is specious, doesn't it? You can't compare across consumers.

When I was taught micro a zillion years ago the theory started with a complete weak ordering of the commodity space, and a formal utility function was something to be derived; the former was enough to get us to Pareto efficiency.

Posted by: Jim Harris on October 10, 2003 12:20 PM>>BTW: If that is somehow your impression, Iīm surely not a conservative

Greinecker, I think you are a theorist. And I'm sure that you're right in saying that optimum holding of goods for some or many of the members in this economy is at the border of the domain.

But I don't think that this does any harm to the differentiabilty, why can't we just take the partial inside-derivative when differentiating at the border?

Posted by: Mats on October 10, 2003 12:52 PMMcClure >>Bill's marginal utility (first derivative) from consuming the first dollar is much higher than mine.>> The first dollar saves you from dying, regardless you are McClure or Bill Gates - no difference there.

>>You can't compare across consumers.>> With the first few dollars, you can indeed compare.

If the dollar that saves Bill Gates from dying has 8 million times the utility of a dollar that saves a Mexican fruit-picker from dying, doesn't that amount to saying that Bil Gates is worth 8 million times more?

Posted by: Zizka on October 10, 2003 03:20 PMMats. I agree. My point here was not to promote the notion that giving me or Bill or anyone else so little income, someone starves. My point was more than attempts to defend unequal income distribution as not being that costly per Social Welfare run into a host of difficult philosophical issues. I may not be Rawlsian on this but my philosophy is close to his. Unequal income distribution that leads to starvation is clearly bad policy.

Posted by: Hal McClure on October 10, 2003 03:31 PMMichael Greinecker writes:

"The neoclassical theory holds up to the highest standards of mathematical rigor."

I assume Michael mean this statement more generally than in just the context of utility theory being discussed here.

Max Sawicky gave reading recommendations over on his blog. One poster commented on Steve Keen's book, saying he found it hard to believe Keen's position that mainstream texts were full of errors. Another said he was glad to see that somebody else, namely Keen, saw the same errors he saw.

I'm of the opinion that neoclassical economics is a logical error.

Let me give an example. In the Arrow-Debreu model of intertemporal equilibrium, initial endowments are taken as given, including of produced means of production. If the initial position is of disequilibrium, firms will quickly adjust the quantities of, at least, circulating capital goods. So the solution paths consistent with the initial data are no longer relevant. Any time to approach an Arrow-Debreu equilibrium is too long.

The above doesn't say there's a mathematical mistake in Debreu's Theory of Value. But it's not an empirical objection or an objection to the "realism" of assumptions, either. It's a kind of an a priori, analytical reason for objecting to neoclassical theory.

And I think one can find many, many other such sorts of objections to neoclassical theory, particularly as presented in textbooks.

And that objection is not original with me. I take it from the Petri reference in footnote 23 in

http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Sraffa3.pdf. Does that essay present a part of neoclassical theory?

By the way, on the subject of interior solutions - I think it an important point in neoclassical economics that what goods are free is determined endogeneously in General Equilibrium models.

Posted by: Robert Vienneau on October 10, 2003 04:20 PM"Actually, one problem with this whole exercise is that the utility functions are most likely ordinal and not cardinal."

Would someone please provide a cite for those definitions of the terms ordinal and cardinal? I'm not familiar with that usage of them.

I'm not sure if I'm missing something--quite likely; it's been decades since I looked at these kinds of questions and I'm ill--but it seems to me that the utility of any given member with respect to the commodity food has a step at starvation level, or at least a very steep edge. So at starvation levels of diet, that partial derivative (my apologies for not using mathematical symbols--I don't think most browsers support MathML) is either undefined, a Dirac delta (if you allow that), or at least so large as to make the mathematics of only theoretical interest--in an actual system, equilibrium would not be established.

In passing, I want to point out that, as telecomm companies learned when developing switching systems for internet traffic, the properties of information systems like economies can be very non-linear. The assumption of continuity in the distribution of internet traffic load turned out to be false; the telecomm companies had to add a great deal of memory to their routers to deal with this. I do not share the neo-classical faith in equilibria, no, not at all. (For cites on this, see http://citeseer.nj.nec.com/cs; use the keywords "self-similar" and "network")

Posted by: Randolph Fritz on October 10, 2003 06:26 PMI cracked the textbooks last night --and OMIGOSH, I had forgotten how much vector calculus you need to know to understand this stuff -and I have forgotten that too!

But I could understand a few things:

-proofs of the existence of equilibrium (and therefore the welfare thoreoms) solve the problem of starvation and zero allocations of various items in the initial endowment by -um, heh-heh, assuming them away. So we start by assuming everyone has enough initial endowment to survive. And you assume rather obscure mathy type assumptions on the maximum number of goods for which you have a zero endowment, so that in equilibrium you will move away from enough zero corners of the initial endowment for an equilibrium to exist.

--Satiation seems to be OK for existence of eqilibrium and the welfare theorems as long as each consumer is not satiated in everything. So if every person wants some more of at least one thing, the math will work out.

--regarding the issue of disequilibrium paths raised by one post - I *think* the best official position is that we just don't know. I was reminded of the Sonnenschein-Mantel-Debreu-etc-etc results, that say, given the observable variables of the system, any time path of observable aggregate goods transacted and transaction prices can be explained as a path converging to equilibrium. So economists just have to make their best guess as to whether we are far away from or moving towards or at equilibrium. You can always cook up some unobservable variables to explain the observations as a path converging to equilibrium (some real economist help me if I have gotten this wrong).

--A few posts raised the issue of integrability. I don't think that is an issue unless you want to try to quanitfy changes in people's welfare in terms of money or a numeraire good. Economists start by assumng an integrable welfare function for each person. But you cannot integrate it in terms of an observable money or numeraire good. That is my understanding of the "integrability problem" (and I may need help on this from a real economist). I just found out that a physicist named Joseph L. McCauley has claimed to have a neat analysis of the integrability problem with solves all of the economists'confusions -but it is very heavy going and I don't understand all of it. But he has a web site where you can get the paper. But be warned -he HATES neoclassical economics.

But on this I am a simple timid fellow and it is enough to know that if my endowment goes away I might have to settle for something way down in my ordering of perferred allocations in society -that is enough to frighten me and I don't inquire further.

--Herbert Scarf has done some work on the problems of economic equilibrium an optimality with out non differntiable and very unsmooth production functions. And he finds problems. So I guess people worried about that should look there and see how it could be adapted to situations where the consumers utility function is non differentiable.

--for stuff on properties of elections and poltical decsion making systems, check out Donald Saari's work.

By the way, I am an applied econometrician, so I think I have an official exemption and am not required to take positions about what the theory people argue about all the time.

I *think* the bottom line of Prof. DeLong's posts is that some of the popular propoganda of free market extremists is very misleading about what society would be like in a pure market economy, and what markets can deliver. The math says something very different from the universal prosperity and happy-happy of market solutions. If there are many people who have more resources than you, they will have much more say in who gets what and how things are done, purely on the basis of initial endowments. Maybe a few impoverished folks will have a mental endowment of genius which will overcome their initial disadvantage in other endowments. But what about the rest?

Any correctoins from a qualified theory person are very welcome.

Posted by: jml on October 10, 2003 06:43 PMthank you, jml--I would not have known where to look, nor spent the time to do the reading.

"proofs of the existence of equilibrium (and therefore the welfare thoreoms) solve the problem of starvation and zero allocations of various items in the initial endowment by -um, heh-heh, assuming them away."

It sounds an awful lot like the neo-classical analysis tacitly assumes a social safety net.

"So economists just have to make their best guess as to whether we are far away from or moving towards or at equilibrium." Ouch! This is like statisticians hoping for that Gaussian central limit.

"Economists start by assumng an integrable welfare function for each person. But you cannot integrate it in terms of an observable money or numeraire good." I think Marx said more-or-less that the basis was time in the control of the person. I don't know if I believe that, agree, or even have got him right.

"Herbert Scarf has done some work on the problems of economic equilibrium an optimality with out non differntiable and very unsmooth production functions." Interesting. I may actually look him up, though I suspect I will not make the time to follow him.

Joe McCauley gets his problem domain from Philip Mirowski, a very interesting economist at Notre Dame. (Due to exercise of hegemonic power, Notre Dame has just denied Mirowski any graduate students to supervise.)

Mirowski shows in More Heat Than Light that neoclassical economics was created by stealing the structure of 19th century mechanics. Call potential energy "utility", kinetic energy "income". Then the integrability conditions state, roughly, that the sum of income and utility is conserved.

The statement is rough because it does not account for the coordinate transformation from commodity space needed to handle the budget constraint. But, still, the integrability or Slutsky conditions are a statement of an (arbitrary) conservation law.

An implication of this is that the utility one receives from consuming a specified consumption bundle does not depend on the order of transactions one uses to convert one's initial endowment into that bundle.

And utility and money are the same sort of stuff, despite what the textbooks say.

Mirowski states that revealed preference theory and the post war turn to topological arguments that don't depend on differentiability did not alter the structure of utility theory enough to remove these properties. But he doesn't go through the math to demonstrate this.

D. Wade Hands has some interesting things to say about the math in his paper, "More Light on Integrability, Symmetry, and Utility as Potential Energy in Mirowski's Critical History", published in an annual HOPE volume of a few years back.

Posted by: Robert Vienneau on October 11, 2003 04:12 AM"Mirowski shows in More Heat Than Light that neoclassical economics was created by stealing the structure of 19th century mechanics."

Oh, interesting. The deeply-embedded assumption of equilibrium of neoclassical economics has always struck me as an import from the physical sciences, since information systems so seldom obey it. Maybe I will make the time to read that.

Web page: http://www.nd.edu/~pmirowsk/

The HTML is a mess, though.

"But I don't think that this does any harm to the differentiabilty, why can't we just take the partial inside-derivative when differentiating at the border?"

The problem in the old way of marginal something equals marginal something is that you get to the border befor you have equalized. I hope this is halway clear.

"Let me give an example. In the Arrow-Debreu model of intertemporal equilibrium, initial endowments are taken as given, including of produced means of production. If the initial position is of disequilibrium, firms will quickly adjust the quantities of, at least, circulating capital goods. So the solution paths consistent with the initial data are no longer relevant. Any time to approach an Arrow-Debreu equilibrium is too long."

Yes, dynamics are big problem in that model. Debreu basically avoided the problem and is very sceptical of the possibility. Here is what Debreu said about the issue in ainterview:

"I thought about those questions of course as everyone must, but it seemed to me that the contributions made were not important. For one thing, when you are out of equilibrium, in economics you cannot assume that every commodity has aunique price because that is already an equilibrium determination. The process should in particular take into account that the same commodity has different prices at the same time so that makes makes dynamics dynamics among other things very hard. And you have to recognize that in classical mechanics you have a simple correlation: force is proportional to acceleration. We have nothing similar in economics. So I have always been very distrustful of dynamics and have mentioned it rarely."

"Would someone please provide a cite for those definitions of the terms ordinal and cardinal? I'm not familiar with that usage of them."

Ordinality means that if u(.) is a utility function and f is a increasing function, then f(u(.)) represents the same utility.Cardinality is more restrictive: If u is a utility function representing cardinal preferences than all other utility functions representing the same preferences are of the form a + b*u, where a is any real number and b is positive.

"So economists just have to make their best guess as to whether we are far away from or moving towards or at equilibrium. You can always cook up some unobservable variables to explain the observations as a path converging to equilibrium (some real economist help me if I have gotten this wrong)."

This is pretty much true if you can look only at the excess demand function. But with more information, one cannot say everything. I donīt know of the relation to dynamics though.

"Mirowski states that revealed preference theory and the post war turn to topological arguments that don't depend on differentiability did not alter the structure of utility theory enough to remove these properties. But he doesn't go through the math to demonstrate this."

I donīt think this is true. The work of Samuelson was extremely physics oriented. He even used physical examples to justify his views in his Nobel lecture. But Debreu was very conscious about that and moving away from it. The corner solution thing I pondered about was actually one of the main differences. The physics copy version of neoclassical economics is the marginal something equals marginal something stuff. Nevetheless, Mirowski is a excellent historian. Together with E.Roy Weintraub and Takshi Negishi he is my favorite history of economic thought person.

Greinecker: "The problem in the old way of marginal something equals marginal something is that you get to the border befor you have equalized."

But we don't have that problem here, we have marginal something times factors like inverted price and weighting. As long as you accept inner derivatives, which I think you do, Brad's exercise should be correct.

Then again, you can always question uniqueness and existence of equilibria. But there seems to be a fairly stable wealth distribution, at least on the time-scales of our everyday lives. And whatever model that would end up showing an optimal equilibrium distribution would anyway implicitly tell that the wealth of the poor is of little importance.

Posted by: Mats on October 12, 2003 09:58 AMThis is in Microeconomic Analysis, 3rd edition, p 331-332. I think it was also in earlier editions, but I don't have any at hand to check. But I certainly didn't invent it---I think it has been known for a long time (although I can't give you a reference.)

Posted by: Hal Varian on October 12, 2003 06:38 PMPerhaps everyone has moved on, but I'll make two notes:

--one post mentions that while just knowing the level of market excess demand is not enough to guarantee covergence to equilibrium, if people know more then there could be convergence. This seems to be true. A mathematician named Smale has shown that if people know the whole excess demand function, or something equivalent, then there is usually convergence to equilibrium. But the math is hard, and I don't fully understand it --I get my info from the discussion of his work by Saari in his latest book, who notes that the informational demands of Smale's mechanism are very very large -and almost certainly unrealistic. And then there are models of stock-flow equilibrium where convergence results are easier to get. I think this work was begun by Clower and Burkinshaw in some very early work on the micro foundations of macro. E. Roy Weintraub disucssues this in one of his books.

--I agree with the posts on the relationship between McCauley's and Mirowski's work. I think the problem with McCauley's work is not that it was inspired by Mirowski's research program -but that McCauley goes from analyzing real-goods economies to analyzing economies with an *important* financial sector without understand recent advances economics has made in that area. But I think McCauley is good to read because he is an expert on the math of integrability and the role it plays in empirical science --so I am reading him and makeing very (very!) slow progress.

I hope someday Prof DeLong will return to this subject and enlighten us on the secret message of his dialogue (I guess I am assuming he is a Straussian now).

Posted by: jml on October 13, 2003 11:16 AM"It sounds an awful lot like the neo-classical analysis tacitly assumes a social safety net."

Itīs more like one has enough to live to beigin with.

"Would someone please provide a cite for those definitions of the terms ordinal and cardinal? I'm not familiar with that usage of them."

If u is a ordinal utility function than f°u represents the same preferences if f is strictly increasing. If u is a cardinal utility function, then all utility function representing the same preferences has the form a + b*u, where a is any real number and b positive.

"Mirowski states that revealed preference theory and the post war turn to topological arguments that don't depend on differentiability did not alter the structure of utility theory enough to remove these properties. But he doesn't go through the math to demonstrate this."

What has remained the same?

Post a comment