October 09, 2003

A Non-Socratic Dialogue on Social Welfare Functions

Glaukon: "Professor!"

Agathon: "Professor! Good to see you. Getting coffee?"

Glaukon: "Yes. I'm teaching. I find that teaching is always and everywhere a caffeine phenomenon."

Agathon: "I tend to find that teaching is usually a bagel phenomenon myself. What are you going to teach them?"

Glaukon: "Social welfare. Utilitarianism. Condorcet. Arrow. Aggregation of preferences. Preference-revealing mechanisms."

Agathon: "Sounds like a full class."

Glaukon: "You have no idea."

Agathon: "Be sure to teach them about the market's social welfare function."

Glaukon: "The market has a social welfare function?"

Agathon: "Under appropriate conditions of perfect competition, non-increasing returns, and the absence of externalities the market's decisions about the production and allocation of goods and services attain a point on the Pareto frontier. Every point on the Pareto frontier maximizes some social welfare function."

Glaukon: "Yes, of course."

Agathon: "Therefore the market, considered as a collective mechanism for making social decisions, chooses to maximize a particular social welfare function. It is instructive to consider what that social welfare function is."

Glaukon: "I resent the tone in which you are talking down to me."

Agathon: "You do not. This part of this conversation never took place in even approximate form in the real world. It is interpolated in order to bring readers of this weblog up to speed. Since I never said my last speech to you, you could not have resented it."

Glaukon: "And I want readers of this weblog to know that I am considerably smarter and more clued-in than he is letting me appear to be."

Agathon: "Are you quite finished?"

Glaukon: "Plato at least worked harder to make his information dumps fit more gracefully into the conversation. I want a better author.

Agathon: "Are you quite finished?"

Glaukon: "Yes."

Agathon: "As I was saying, the market system chooses an allocation. That allocation can only be justified under the assumption that moves along the Pareto frontier in every direction--moves that transfer wealth from one member of society to another--are of no benefit to social welfare, while moves toward the Pareto frontier do benefit social welfare. If we restrict ourselves to social welfare functions that are weighted sums of individual utilities, that means that the market system's social welfare function gives each individual a weight inversely proportional to his or her marginal utility of wealth."

Glaukon: "Didn't somebody say about society that there was no such..."

Agathon: "Hush! If you want to quote Margaret Thatcher, you must introduce her as a speaking character in this dialogue and grant her some of her time..."

Glaukon: "I? You're the authorial stand-in in this dialogue, not me..."

Agathon: "That means that the market system, in weighting utilities and adding them up, gives you a much lower utility than it gives Richard Cheney. In fact, if marginal utility of wealth is inversely proportional to the square of lifetime wealth, the market system gives Richard Cheney about 400 times as big a weight as it gives you."

Glaukon: "That's sick."

Agathon: "And it gives Bill Gates a weight about 400,000,000 times as big a weight as it gives you."

Glaukon: "That's sicker."

Agathon: "But it gives you about 40,000 times the weight it gives your average Bengali peasant, who thus has about 1/16,000,000,000,000 the amount of the market system's concern as Bill Gates has. Will you teach that?"

Glaukon: "They'll call me a Communist!"

Agathon: "But it's true!"

Glaukon: "That I'm a Communist?"

Agathon: "No. That that's what the market system does!"

Glaukon: "We are value neutral economists! We don't care about distribution! We care about efficiency!"

Agathon: "But claiming that you don't care about distribution is implicitly saying that shifts in distribution are of no account--which can be true only if the social welfare function gives everybody a weight inversely proportional to their marginal utility of wealth."

Glaukon: "You're introducing politics into a value-neutral technocratic social science."

Agathon: "Politics?! Moi? I'm simply evaluating the derivatives of a social welfare function under the assumption that the market allocation is its ArgMax. What could be more technocratic than that? I'm just trying to attain a little clarity of thought."

Thrasymachus: "But where rule rests not--as somebody or other said at one of Old Joseph de Maistre's little soirees in St. Petersburg--on the hangman, but on misdirection and confusion, to strip away the veils of alienation and false consciousness that keep humans from perceiving their species-being, the act of unveiling is itself a powerfully political act."

Agathon: "Are you Thrasymachus or Karl Marx?"

Thrasymachus: "Ah. Marx thought unveiling was a good thing. I think it is neither good nor bad, for 'good' like 'justice' is really just another word for the interest of the stronger party."

Glaukon: "And we gave you tenure here at Berkeley?"

Thrasymachus: "Shhh! The humanities departments still think relativism is sexy. They haven't yet figured out that to assume a position of relativism--like the claim to be neutral on issues of distribution--is really a statement that you are on the side of the powerful."

Agathon: "And are you?"

Thrasymachus: "It is the just and the good--or, rather, the 'just' and the 'good'--thing to do.

Posted by DeLong at October 9, 2003 03:00 PM | TrackBack

Comments

Today, I love you, Brad.

Posted by: Keith M Ellis on October 8, 2003 08:20 PM

That was truly great!

Posted by: Hannibal on October 8, 2003 08:24 PM

Amused and enlightened in equal parts.

I don't remember the last time that happened.

Posted by: P6 on October 8, 2003 09:04 PM

When did Thrasymachus get tenure? Does Peter Berkowitz know about this?

Posted by: Invisible Adjunct on October 8, 2003 09:05 PM

It must have been a whole wheat bagel.

Posted by: northernLights on October 8, 2003 09:07 PM

Oh, the Sophists *always* get tenure. The Socratics...well, they get the hemlock. You know how it is.

Posted by: Keith M Ellis on October 8, 2003 09:09 PM

Very nice dialogue. Thanks. I'd like to hear more. All of this is a long time ago for me now, but I think Martin Shubik showed that the market social welfare function could be implemented by an election for allocations with each particpant's vote is weighted according to their wealth. So does this mean that the market social welfare function suffers from the limitations of Arrow's theorm and subject to manipulation (Gibbard-Satterthwaite results)? And so should be condemned as flawed like political elections? But then some one (I think Chichilnitsky and maybe someone else) showed that if and only if everyone's preferences for everything are nicely concave like in the Micro 101 figures, then we live in a single peaked world and the market social welfare function escapes condemnation of not fufilling all of Arrow's requirements? So does are faith in the market social welfare function over political elections rest on our faith in unobservable Econ 101 preference functions for everyone and everything all the time?

And what about each person's wealth that creates the weights? Do we have to believe in dynastic families (all the the way back to Ur) who incorporated each of their children's discounted future welfare? (Looks like some of those founders blew it big time). Or is it OK to posit that the older generation can choose and the arriving generation is forced to let bygones be bygones and we start afresh each day? (the current wealth distribution is the current wealth distribution and don't complain!)

And what is the weight of market participants born in the future? If it is zero, what does that mean for the market social welfare function? Or must future generations put their faith in the current dynasts (the current generation) to weight their welfare properly? But how is that implemented in the real world? Where are the markets?

If I've gotten this mixed up, I apologize for wasting electrons. But if I haven't I'd like to hear the dialogue continue.

Posted by: jml on October 8, 2003 10:01 PM

I would really appreciate some explanation of the intuition behind this result.

Posted by: David J. Balan on October 8, 2003 10:27 PM

I trust you realize that your institution now owes David Horowitz an unfunny objectivist with tenure.

Posted by: julia on October 8, 2003 10:54 PM

Very enlightening post...but one query: Isn't the specification of a social welfare function a subjective task? It seems that the dialogue presupposes a certain functional form for the social welfare function but how would this dialogue be different if a different social welfare function were introduced?

And why is this the preferred social welfare function? Just curious...

Posted by: EcoDude on October 9, 2003 05:09 AM

Brad, I think your analysis suffers a fatal flaw, and here it is:

"If we restrict ourselves to social welfare functions that are weighted sums of individual utilities, that means that the market system's social welfare function gives each individual a weight inversely proportional to his or her marginal utility of wealth.'

True.

"That means that the market system, in weighting utilities and adding them up, gives you a much lower utility than it gives Richard Cheney."

False, or more accurately, only true with zero probability. The inversely proportional weighting based on MU of wealth leads to this greater weighting of Cheney's utility function ONLY if Cheney and everybody else has the EXACT same utility function in income. If Mr. Cheney has a utility function in income that starts super high and falls slowly, and I have a utility function in income that starts less high and falls quickly, then Mr. Cheney and I could have exactly the same marginal utility in income even when Mr. Cheney's income is a lot higher, so that the market's social welfare function does not necessarily weight the richer person's utility more than the poorer person's.

In short, Brad, you (unknowingly, I think) snuck the assumption of homogeneous agents into your analysis. But the 1st Welfare theorem does not require homogeneous agents in order to work.

However, I still think your analysis is interesting. If we believe that agents have similar utility functions in income or even if we believe that people's utility functions in income are uncorrelated with their actual income, then your analysis still applies.

But it could be all Greek to me.

Posted by: Keith on October 9, 2003 05:19 AM

JML --

I thought that Arrow's impossibility theorem applied when preferences were strictly ordinal. A market function -- even if it were done by voting -- involves (or allows for) intensity of preferences (which I think is a reasonable assumption), so I'm not sure the theorem would apply.

Along these exact lines, does anyone know if Arrow's theorem applies to a voting system like the one for MVP in baseball? In that system, you can vote for ten candidates. 1st place gets 14 points, and then 2nd place gets nine, all the way down to 10th place gets one. You don't (I think) have to vote for all ten candidates, either.

Posted by: James Surowiecki on October 9, 2003 05:53 AM

Rather than clarity, I feel you've created a muddle! Societies have a large number of different benefits, burdens, duties and responsibilities to distribute among the members, and some of these differ in kind. By lumping them all together under one rubric (social welfare), and claiming that these are to be distributed by the same mechanism (market allocation),you ignore that we have come together as a society agreeing to distribute different benefits, burdens, duties and responsibilities in different ways, based on their differing kinds.

To give just one example, we originally contracted to distribute the benefits of political power in an egalitarian way (one citizen - one vote, anyone can run for office, etc), even while we allow the distribution of property and its benefits to be allocated via some market mechanism.

That there is "leakage" between benefit kinds and their distribution methods (ie, one can buy a govonorship, or use political power to increase property holdings) is a problem of weak/poorly enforced laws, and straying from a poorly seen ideal.

Also, by focusing everything into one muddled category (social welfare), you ignore that society's duties and responsibilities will also be distributed according to "market allocation". So, by your theory, we should also be distributing duties and responsibilities by market allocation systems, and people like Gates or Cheney ought to be assigned the most difficult duties, too, like curing malaria, aiding the handicapped, defending the country from enemies, and relieving other "natural burdens" and carrying out other of society's (sometimes onerous) duties.

There is so much more to be said here, but to try to summarize, our "original contract" in this society is NOT to distribute every one of society's benefits, burdens, duties and responsibilities by some idealized market mechanism. Rather, we agreed distinguish benefits, burdens, etc by their *kinds* and use different mechanisms of distribution for different kinds.

tjallen

Posted by: tjallen on October 9, 2003 06:52 AM

>>In short, Brad, you (unknowingly, I think) snuck the assumption of homogeneous agents into your analysis. But the 1st Welfare theorem does not require homogeneous agents in order to work. <<

Yep. You caught me.

:-)

Posted by: Brad DeLong on October 9, 2003 07:27 AM

I thought Agathon's argument broke down over the requirement of perfect competition. Because if there were perfect competition, no computer user would have purchased a Microsoft product in the last ten years, and Bill Gates would be standing on a virtual street corner selling cyber-pencils.

Posted by: Squeech on October 9, 2003 07:47 AM

If wealth has diminishing marginal utility for individuals, as we assume everything else does, then what happens? For any single individual, his first $10,000 will have much more untility than his ten thousandth $10,000. So in order for an additional $10,000 (with its lesser utility, piled on top of $40 or so billion) in Bill Gates' pocket to have more utility than the first or second $10,000 in a poor man's pocket, his desire for wealth must be virtually infinite.

Posted by: Zizka on October 9, 2003 07:48 AM

Zizka: You would deny that Bill Gates' desire for wealth is virtually infinite?

Seriously, I would guess that individuals' utility function for monetarily acquirable wealth is correlated with wealth. People who value wealth and buyable comforts will work harder, and take high paying jobs which are boring/annoying. People who value quality of life more will take "easier" lower paying jobs with shorter hours, and spend more time with the family. Bill Gates worked long hours to start Microsoft to hit the jackpot of wealth. Brad DeLong worked long hours in grad school to hit the jackpot of quality of life - a job in the higher reaches of Academia. We can assume from their relative choices that, ceteris paribus, Bill values wealth more than Brad. Of course, ceteris is not paribus, and I'd be shocked if the marginal utility of an additional 10 grand to Bill is anywhere close to the utility of that money to Brad. (Maybe not, if Brad is satiated in his demand for goods and services, but I doubt that he makes THAT much at a state university.)

Posted by: rvman on October 9, 2003 08:03 AM

''Agathon: "Under appropriate conditions of perfect competition, non-increasing returns, and the absence of externalities the market's decisions about the production and allocation of goods and services attain a point on the Pareto frontier. Every point on the Pareto frontier maximizes some social welfare function."

Glaukon: "Yes, of course."''

Of course? Why is the market supposed to pick equilibrium prices? That doesn´t follow in any way from the assumptions.

''Very enlightening post...but one query: Isn't the specification of a social welfare function a subjective task? It seems that the dialogue presupposes a certain functional form for the social welfare function but how would this dialogue be different if a different social welfare function were introduced?''

The maximum of a linear combination of utility functions (which gives a SWF), defined on all feasible allocations is always pareto optimal. But information on marginal utility usually gets lost.

''I thought that Arrow's impossibility theorem applied when preferences were strictly ordinal. A market function -- even if it were done by voting -- involves (or allows for) intensity of preferences (which I think is a reasonable assumption), so I'm not sure the theorem would apply.''

No. The real reason why Arrow´s theorem poses no problem is the assumption of no externalities, which goes against the unrestricted domain assumption in arrows theorem. But it is true that Arrows theorem depends on the ordinality of preferences. Apart from that axiom, utilitarianism would satify all other axioms.

Posted by: Michael Greinecker on October 9, 2003 10:42 AM

And of course, selfless people who care nothing about money will choose to have a series of $7/ hr. temp jobs. It works out so beautifully.

Posted by: Zizka on October 9, 2003 11:21 AM

And of course, selfless people who care nothing about money will choose to have a series of $7/ hr. temp jobs. It works out so beautifully.

Posted by: Zizka on October 9, 2003 11:26 AM

1) What ever became of John Hicks' neo-pareto criterion, that an exchange was optimal, if the winning party of an exchange could compensate the losing party to the exchange from his gains and both would be at least as well off as before? Was this purely an analytic device or an appeal to institutional imagination, as well?

2) The best general definition of specifically economic action I ever came across was that economic action involved the searching out, realization, and distribution of surpluses. (I believe this was in an article by Maurice Allais in a collection called "The Foundations of Economics", Baranzini and Scazzierei, eds.) This has a number of advantages over the marginalist utility-preference function maximalisation model as a generalization. A) It applies equally across all 3 domains of economics: production, exchange and consumption. B) It applies to all possible economies that are not primitive subsistence economies, i.e. that produce a surplus product from the portion of the total product that is consumed in the process of its production. C) It accounts plausibly for the efficiency of markets in that any market with a sufficient number of agents subscribing to it would squeeze out all available surpluses. And it does so precisely by appealing to the actions of agents rather than the autonomous functioning of markets. Furthermore, it could account for possible inefficiencies of markets, insofar as e.g. they were overly specialized or restricted in access due to economies of scale. D) It makes no appeal to preferences that are either tautological or unknowable, but concerns only what is producible and available at any given time. E) It does not specify agents, as to whether they are individual or collective (e.g. organizations), nor does it specify their activities, as to whether they are competitive or cooperative in nature. F) It allows for the specification of base-line constraints, such as wage rates not exceeding the actual productivity of labor or returns to capital not being less than the replacement costs of capital, without predetermining the middle ranges. In other words, it follows an epistemically negative, hence fallibillistic approach rather asserting unprovably that any given arrangement is necessarily optimal. G) By specifying economic action, it allows for a consideration of the economic dimension of orientations to purposes that are primarily not economic in nature, without reducing possibly incommensurable values to a single continuous notion of utility.

Posted by: john c. halasz on October 9, 2003 12:22 PM

OK, I promise to brush up on this subject before I post again. But for now I am pretty sure of the following:

To Michael Greinecker and James Surowiecki--
You can get around Arrow's theorem if societies preferences are single peaked. That means that individuals' preferences follow a certain pattern across the alternatives. For example the concave indifference curves you see in Econ 101 result in single peaked preferences. Whether you suppose that preferences are ordinal or cardinal makes no difference.
I don't think that introducing intensity of preferences helps much -you still violate one of the requirements for a *good* method. The requirement of "independence of irrelevatn alternatives" (the requirement that you can all information about a person's preferences form looking at preferences between pairs of alternatives, and only looking at that) is violated. But if you can live with a weaker assumption (that you can look at several choices between pairs of alternatives and check that all individuals' preferences are transitive) then introducing intensity of preferences gets you a good system by Arrow's criteria. I think Samuelson and Saari have verified this.

On the issue of homogenous agents and diminishing marginal utility of consumption, I don't see how that affects the ethical issue, at least in a from a neo-classical perspective. We are not allowed to compare different individuals utilities, regardless of whether they are homogenous or not. And isn't diminishing marginal utility incompatible with the assumption that people's preferences are ordinal?

The bottom line is that in the neo-classical world, people can rank all the possible allocations of of goods and work. And if a person has little wealth, they don't have much say in which allocation is chosen for society by the market mechanism.

Posted by: jml on October 9, 2003 12:35 PM

I may sound a bit dated here, given that nothing I'm going to say hasn't been said 80 years ago, but... I object to Keith's objection (unwisely, I know, since Brad has already accepted it, but oh well.).

What is marginal utility of wealth anyway? Is utility something we actually think exists, or is it just a way of expressing monotonic preferences in a way that is subject to the nicities of being a continuous function, so that any ol' function will do so long as the relations of preferences will be the same? If the latter, does talking about the actual magnitude of utility functions even make sense, since one could just as well say that Bill Gates's utility of wealth for the first $10,000 is one billionth of mine, and so long as his utility for other things like, say, leisure, memorizing poetry, etc, are similarly small, his preferences would come out the same as if he had utility like mine in magnitude and priorities like his. So, really, the magnitude of the utility function would just be another way of putting weights in, another way for our capitalist oppressors to claim that the market is generating optimal results according the the SWF that weights people inverseley proportional to their marginal utility of wealth (does it have a name?) without claiming to be weighting the rich more than the poor.

And if utility IS a real thing from the world, not just a way of making monotonic, ordinal preferences subject to calculus, then... well... then I suppose Keith's objection is right, but isn't that view a bit weird?

Posted by: Julian Elson on October 9, 2003 12:53 PM

john c. halasz:

1. The (Kaldor-)Hicks-criterion is widely used in applied economics. But it faces a huge problem:

It isn´t always transitive.

2. This reminds me on a paper Hayek wrote, you ma like it: http://www.mises.org/journals/qjae/pdf/qjae5_3_3.pdf

jml:

It is correct that the independence of irrelevant alternatives rules the conditions for utilitarianism out. That why I said that it depends on preferences being ordinal. I think it was Amartya Sen who clarified the relation between Arrow´s theorem and inter-individual utility comparisons.

''And if utility IS a real thing from the world, not just a way of making monotonic, ordinal preferences subject to calculus, then... well... then I suppose Keith's objection is right, but isn't that view a bit weird?''

Julian Elson:

Well, one can get a empirically meaningful interpretation of marginal utilities if we use preferences on lotteries. Given some further rationality criteria for decision-making under uncertainty, preferences can be represented by a utility function u and all other utility functions representing the same preferences have the form a + b*u where a is any real number and b is positive. In this context, marginal utility makes sense. The idea goes like this: We assume your "utility" for a dollar sinks with every extra dollar you make. Now you wouldn´t accept a bet in which you can win a dollar with 50% probability and lose a dollar with 50% probability. So you require a higher prize when you win to play this.

Here is something on it: http://philosophy.wisc.edu/hausman/524/expected_utility_theory.htm

Now homogeneous agents are basically simply agents that value a dollar independently of how many they already have. If you double all they have they would still have the same marginal utility for everything.

Posted by: Michael Greinecker on October 9, 2003 01:35 PM

john c. halasz:

1. The (Kaldor-)Hicks-criterion is widely used in applied economics. But it faces a huge problem:

It isn´t always transitive.

2. This reminds me on a paper Hayek wrote, you ma like it: http://www.mises.org/journals/qjae/pdf/qjae5_3_3.pdf

jml:

It is correct that the independence of irrelevant alternatives rules the conditions for utilitarianism out. That why I said that it depends on preferences being ordinal. I think it was Amartya Sen who clarified the relation between Arrow´s theorem and inter-individual utility comparisons.

''And if utility IS a real thing from the world, not just a way of making monotonic, ordinal preferences subject to calculus, then... well... then I suppose Keith's objection is right, but isn't that view a bit weird?''

Julian Elson:

Well, one can get a empirically meaningful interpretation of marginal utilities if we use preferences on lotteries. Given some further rationality criteria for decision-making under uncertainty, preferences can be represented by a utility function u and all other utility functions representing the same preferences have the form a + b*u where a is any real number and b is positive. In this context, marginal utility makes sense. The idea goes like this: We assume your "utility" for a dollar sinks with every extra dollar you make. Now you wouldn´t accept a bet in which you can win a dollar with 50% probability and lose a dollar with 50% probability. So you require a higher prize when you win to play this.

Here is something on it: http://philosophy.wisc.edu/hausman/524/expected_utility_theory.htm

Now homogeneous agents are basically simply agents that value a dollar independently of how many they already have. If you double all they have they would still have the same marginal utility for everything.

Posted by: Michael Greinecker on October 9, 2003 01:40 PM

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

James: Arrow's theorem still applies to the kind of voting in MVP balloting. It's not hard to imagine a situation where you can game the results. Say you think player A should be the MVP and you think player B is the second-best player, but that you think other voters will undeservedly reward him with the MVP. You can influence the outcome by giving player B a lower ranking that you would otherwise.

Posted by: Walt Pohl on October 9, 2003 02:16 PM

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

James: Arrow's theorem still applies to the kind of voting in MVP balloting. It's not hard to imagine a situation where you can game the results. Say you think player A should be the MVP and you think player B is the second-best player, but that you think other voters will undeservedly reward him with the MVP. You can influence the outcome by giving player B a lower ranking that you would otherwise.

Posted by: Walt Pohl on October 9, 2003 02:21 PM

Walt --

I don't think the gaming question and Arrow's theorem are necessarily connected. Arrow assumes sincere voting, and then says when there are more than two alternatives, no voting scheme satisfies these minimum conditions. I can see that the MVP voting can be gamed, but I think -- as jml said -- that if you accept a slightly weaker version of the independence assumption, then it satisfies all of Arrow's conditions.

Posted by: James Surowiecki on October 9, 2003 02:44 PM

Oh, we claim utility is ordinal (order matters, size doesn't) but we treat it as cardinal (size matters) all the time. "Expected" Utility, marginal utility, social and other welfare analyses. I would argue there is some underlying "utility" but that it is impossible to actually "functionalize" it accurately for any specific individual, let alone groups.

Zizka: I'm not claiming that people take $7/hr. jobs rather than becoming Bill Gates out of selflessness. They ARE making an economic choice when they choose between a fast food or retail job at $6.00 and a job scrubbing toilets or roofing at $10. Same skill set, same required experience. Just different levels of work/onerousness. The person who takes the latter values money more relative to comfort (and utility really is relative) than the person who picks the former.

Posted by: rvman on October 9, 2003 03:09 PM

These comments are all very interesting. I'm going to have to crack a few textbooks tonight -especially on the interpretation of the "marginal value of wealth" that determines the weights. (And by the way, I am thinking of this in terms of a real goods economy with no fiat currencies)

Will there be another non-Socratic dialogue to enlighten us soon?

Posted by: jml on October 9, 2003 03:46 PM

I remember being taught long ago that economics had decided to forbid any attempt to measure or define utility, or to compare the one person's utility in any way with another's. This was glibly explained to me as being because of the technical difficulty of quantifying "utils", partly because of differences of preferences. Later on someone else said that it was in order to avoid the egalitarian aspects of utilitarianism, which made sense to me.

If Bill Gates's four-millionth $10,000 has the same utility as a Mexican fruit-picker's first $10,000 (let's say, a utility of one), what would the utility of Gates's first $10,000 have been? Is there a way of putting a number on it?

Posted by: Zizka on October 9, 2003 04:56 PM

Hmm... if all agents have the same marginal utility of wealth no matter what, then wouldn't that mean all agents would be weighted the same for the optimal-market SWF to work?

Posted by: Julian Elson on October 9, 2003 05:14 PM

The comments are TOO interesting, and I'm NOT going to crack open no textbooks this morning.

Brad, it was great! Not completely fair, technically, but really cute.

Maybe I missed somewhere your response about how the 80s raiders were a free(ish) market response to excessive CEO salary, but Congressional support for the "poor" exec victims stopped that correction -- so we have obscene management ripoffs, now.

Posted by: Tom Grey on October 9, 2003 05:21 PM

>>If Bill Gates's four-millionth $10,000 has the same utility as a Mexican fruit-picker's first $10,000 (let's say, a utility of one), what would the utility of Gates's first $10,000 have been? Is there a way of putting a number on it?<<

One often-used utility function (because it doesn't feel very wrong, and because it is easy to calculate with) is log utility: your utility is proportional to the log of income, which means that the marginal utility of your last dollar is inversely proportional to your income.

If Bill Gates and the guy from Oaxaca both have utility that is equal to a constant times the log of their income, then if Bill Gates's $40,000,000,000th dollar has the same marginal utility as the guy from Oaxaca's $10,000th, then Bill Gates's 10,000th dollar had...

4,000,000 times the utility to Gates than the guy from Oaxaca's dollar had.

Posted by: Brad DeLong on October 9, 2003 05:44 PM

rvman - you are really taking revealed preference to unwarranted extremes here. Next thing, you'll be telling me I have chosen to be 43 years old.

Posted by: Tom Slee on October 9, 2003 06:13 PM

Now that it's been properly quantified I feel much better.

Posted by: Zizka on October 9, 2003 06:40 PM

Now that it's been properly quantified I feel much better.

Posted by: Zizka on October 9, 2003 06:45 PM

"Oh, we claim utility is ordinal (order matters, size doesn't) but we treat it as cardinal (size matters) all the time. "Expected" Utility, marginal utility, social and other welfare analyses. I would argue there is some underlying "utility" but that it is impossible to actually "functionalize" it accurately for any specific individual, let alone groups."

With choice under uncertainty, one could principally test the cardinal utility of people. It´s an important issue in subjectivist theories of probability. Leonard Savage wrote some interesting stuff on the issue.

Posted by: Michael Greinecker on October 9, 2003 10:43 PM

Zizka:

I recall an account of a Bill Gates put-down of a competitor as having "finite greed". Can't find it in a Google search, though. Probably apocryphal, but seems to capture the essence, doesn't it?

Posted by: Curt Wilson on October 9, 2003 11:38 PM

Curt: I never hoped to get such a neat answer to my question. Perhaps Gates has a genetic difference so that his marginal utility of wealth stays the same or even increases.

Actually at some point wealth in terms of utility or consumption ("use") becomes irrelevant, since you have two of everything and only so much time and further wealth only adds to your choices ("Should I take Maria Callas of Jacqueline Kennedy out on on which of my yachts?"). So power and fame are the motivators. Economics tries to count these as "consumption preferences" I suppose.

Posted by: Zizka on October 10, 2003 03:33 AM

I think Ted Turner once came up with the idea that everyone of the 100 richest folks or so worldwide spends the same huge amount for charity, so that they can help but still have the same status ranking...

Posted by: Michael Greinecker on October 10, 2003 07:46 AM

I think Ted Turner once came up with the idea that everyone of the 100 richest folks or so worldwide spends the same huge amount for charity, so that they can help but still have the same status ranking...

Posted by: Michael Greinecker on October 10, 2003 07:51 AM

All in all, a splendid exhibition of what happens when you employ minute-analysis of CPA quantifiable kind to such massive ideas as [for just One] 'Quality', and do not notice when

humanity blanched, dissolved Arrow's Theorem in fuming nitric acid.. departed the melee and left on a desert trek, with head shaking..

Thanks, folks - and yes I (hope I) discern the wry mea culpas within these fine excoriations. WTF - keeps us off the streets, no?


Ashton, no doubt <1 µMUW == mu == µµ
(a lover of involute recursive Absences-of-Plans which are Fun.. enough, despite the futility.)


Posted by: Ashton on October 10, 2003 04:04 PM

All in all, a splendid exhibition of what happens when you employ minute-analysis of CPA quantifiable kind to such massive ideas as [for just One] 'Quality', and do not notice when

humanity blanched, dissolved Arrow's Theorem in fuming nitric acid.. departed the melee and left on a desert trek, with head shaking..

Thanks, folks - and yes I (hope I) discern the wry mea culpas within these fine excoriations. WTF - keeps us off the streets, no?


Ashton, no doubt <1 µMUW == mu == µµ
(a lover of involute recursive Absences-of-Plans which are Fun.. enough, despite the futility.)


Posted by: Ashton on October 10, 2003 04:06 PM

All in all, a splendid exhibition of what happens when you employ minute-analysis of CPA quantifiable kind to such massive ideas as [for just One] 'Quality', and do not notice when

humanity blanched, dissolved Arrow's Theorem in fuming nitric acid.. departed the melee and left on a desert trek, with head shaking..

Thanks, folks - and yes I (hope I) discern the wry mea culpas within these fine excoriations. WTF - keeps us off the streets, no?


Ashton, no doubt <1 µMUW == mu == µµ
(a lover of involute recursive Absences-of-Plans which are Fun.. enough, despite the futility.)


Posted by: Ashton on October 10, 2003 04:09 PM

Apologies - the board failed to update upon 'posting', never completed the action. So I tried x-times with various junk filters turned off.

Perhaps someone can delete the tedious repetition (?)

Ashton

Posted by: Ashton on October 10, 2003 04:20 PM

As a Greek man, I believe that this..episode is appropriate for film or tarining serial on TV!!

Marvellous, professor DeLong

Posted by: Ilias Karavolias on October 14, 2003 06:48 AM

So here's how this might work out in practice. If you have a bunch of nice paintings decorating your walls, and Bill Gates came to visit, and he took a liking to your paintings, you would probably both find that a mutual exchange was possible in which he would end up with all of the paintings and you would end up with a pocket full of money, and you'd both be happier.

This is because Bill Gates' marginal utility of money is, we will suppose, less than yours. That is, he can spend thousands of dollars without much concern, whereas for you to receive thousands of dollars is of great value. In this way, people who have less marginal utility of money, that is, people who spend money easily, will tend to end up with more goods than people who want or need to hold money and can't spend it easily.

Moving towards the Pareto frontier, we engage in mutually agreeable trades like this, each side becoming better off than before. Eventually no more such trades are possible, everything is a wash, and we are at the frontier. We can move sideways along the frontier only by exchanging goods of equal value, and that includes money. At that point, if money is worth N times more to me than it is to you (because you are much richer than me), I will have sold you my goods until what I have left is worth N times more to me than it is to you, just like the money is. That is, for every good, the ratio of our marginal utilities is equal to the ratio of our marginal utilities of money, at the Pareto frontier.

I don't think it follows from this that the total allocations are proportional to these ratios, not without additional assumptions about the shape of the diminishing returns curve. But qualitatively it is clear that the ones who spend more money get more goods, in a purely voluntary trading regime. To keep this from happening you must prevent people from making mutually agreeable trades that make both of them happier.

Posted by: Cypherpunk on October 14, 2003 07:55 PM

So here's how this might work out in practice. If you have a bunch of nice paintings decorating your walls, and Bill Gates came to visit, and he took a liking to your paintings, you would probably both find that a mutual exchange was possible in which he would end up with all of the paintings and you would end up with a pocket full of money, and you'd both be happier.

This is because Bill Gates' marginal utility of money is, we will suppose, less than yours. That is, he can spend thousands of dollars without much concern, whereas for you to receive thousands of dollars is of great value. In this way, people who have less marginal utility of money, that is, people who spend money easily, will tend to end up with more goods than people who want or need to hold money and can't spend it easily.

Moving towards the Pareto frontier, we engage in mutually agreeable trades like this, each side becoming better off than before. Eventually no more such trades are possible, everything is a wash, and we are at the frontier. We can move sideways along the frontier only by exchanging goods of equal value, and that includes money. At that point, if money is worth N times more to me than it is to you (because you are much richer than me), I will have sold you my goods until what I have left is worth N times more to me than it is to you, just like the money is. That is, for every good, the ratio of our marginal utilities is equal to the ratio of our marginal utilities of money, at the Pareto frontier.

I don't think it follows from this that the total allocations are proportional to these ratios, not without additional assumptions about the shape of the diminishing returns curve. But qualitatively it is clear that the ones who spend more money get more goods, in a purely voluntary trading regime. To keep this from happening you must prevent people from making mutually agreeable trades that make both of them happier.

Posted by: Cypherpunk on October 14, 2003 07:56 PM

[Apologies if this is a duplicate. I waited 15 minutes after the "Post" command timed out in an apparent failure but the message didn't appear on the page. So I'm trying again.]

So here's how this might work out in practice. If you have a bunch of nice paintings decorating your walls, and Bill Gates came to visit, and he took a liking to your paintings, you would probably both find that a mutual exchange was possible in which he would end up with all of the paintings and you would end up with a pocket full of money, and you'd both be happier.

This is because Bill Gates' marginal utility of money is, we will suppose, less than yours. That is, he can spend thousands of dollars without much concern, whereas for you to receive thousands of dollars is of great value. In this way, people who have less marginal utility of money, that is, people who spend money easily, will tend to end up with more goods than people who want or need to hold money and can't spend it easily.

Moving towards the Pareto frontier, we engage in mutually agreeable trades like this, each side becoming better off than before. Eventually no more such trades are possible, everything is a wash, and we are at the frontier. We can move sideways along the frontier only by exchanging goods of equal value, and that includes money. At that point, if money is worth N times more to me than it is to you (because you are much richer than me), I will have sold you my goods until what I have left is worth N times more to me than it is to you, just like the money is. That is, for every good, the ratio of our marginal utilities is equal to the ratio of our marginal utilities of money, at the Pareto frontier.

I don't think it follows from this that the total allocations are proportional to these ratios, not without additional assumptions about the shape of the diminishing returns curve. But qualitatively it is clear that the ones who spend more money get more goods, in a purely voluntary trading regime. To keep this from happening you must prevent people from making mutually agreeable trades that make both of them happier.

Posted by: Cypherpunk on October 14, 2003 08:19 PM
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