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Utility/Wealth Maximization
By Brad Rodgers on Sunday, January 06, 2002 - 03:47 am:

Suppose that there is a function f(x) such that f'(x)<1, f''(x)<0 (it's concave), and f(0)=0, and suppose that there are numbers a1 through an such that a1+a2+...+an-1+an=1. Does the maximum of

G(x) = Sn k=1f(ak)

Occur at a1=a2=...=an=1/n?

I'm pretty sure it does; however, I'm not sure I can prove it. If someone could, the application would be obvious, as the function desribed is a fairly obvious representation for the amount of happiness derived from a given amount of wealth for some person. If that were true, and, this equation were true, (which both probably are), it would be a fairly rigorous argument for equal distribution of wealth bringing the greatest amount of happiness...

Brad

By Dan Goodman on Sunday, January 06, 2002 - 05:54 am:

OK, I think I have a proof of this (and then some political discussion at the end :-).

We use induction, n=1 is obvious enough. Also, we use the fact (necessary to make it even true) that f'(x)>=0. (I don't think we need f'(x)<1.)

We want to somehow reduce case n+1 to case n. To maximise f(a1)+...+f(an+1) s.t. a1+...+an+1=1 we want to maximise over all an+1 what we get when we maximise f(a1)+...+f(an)+f(an+1) (the last term is constant when varying only the a1 to an+1) s.t. a1+...+an=1-an+1. So we have two maximisations going on, let's fix an+1 and do the second maximisation.

Basically, to maximise f(a1)+...+f(an+1) we need only maximise f(a1)+...+f(an) because the last term is constant. If we let bi=ai/(1-an+1) and g(x)=f((1-an+1)x) then the problem is to maximise g(b1)+...+g(bn) s.t. b1+...+bn=1. Since g satisfies all the conditions that f does, we have (using the inductive hypothesis) that this occurs when b1=...=bn=1/n. In other words, ai=(1-an+1)/n.

So, now let's do the first maximisation. We have that f(a1)+...+f(an)=nf((1-an+1)/n). So we want to maximise h(x)=nf((1-x)/n)+f(x) for 0<=x<1. (Replacing an+1 with x.) Now h'(x)=-f'((1-x)/n)+f'(x). This achieves a max when h'(x)=0 so f'((1-x)/n)=f'(x). Since f'(x) is a strictly decreasing function this means that (1-x)/n=x. In other words x=1/(n+1). So, quick algebra, ai=1/(n+1) for all i=1,...,n+1.

Hope that proof is OK (you need an additional argument to exclude the possibility that any of the ai are 0 or 1, if you can do this it deals with the problem that the bi might not be defined and that the maximum of h(x) might not correspond to a point where h'(x)=0), I haven't checked it over.

Right, now the politics...

Firstly, I'm pleased to see this sort of socialistic thinking going on :-)

Secondly, it unfortunately wouldn't provide a rigorous argument for equal distribution of wealth bringing the greatest amount of happiness. The reason is the assumption that a1+...+an=1. An economist would argue that the distribution of wealth will effect the total amount of wealth.

For example, it might be that societies where everyone is equal cannot achieve economic growth because there is no incentive for working hard. In other words, if everyone gets equal money whatever they do, why should they work rather than just sitting at home watching TV? So, although an equal distribution of wealth might bring greatest maximum happiness it might be that a policy of equal wealth will diminish maximum happiness in the long term because of the social effects of the policy.

Free market capitalist apologists will often use an argument like this to dismiss egalitarian systems of society. As a strict objection to a system that has an enforced equality of wealth it has some credence. However, it doesn't address the possibilities of societies that do not enforce equality but are naturally egalitarian. These can have incentives to work just as a free market economy can. (For example, reward according to effort.)

Thirdly, is maximising f(a1)+...+f(an) the best thing to do? For example, is a society with 99 people with utility 1 and 1 person with utility 901 (total utility 1000) better than a society where everyone has utility 9 (total utility only 900)? There are lots of different things we could maximise (these are loosely called social welfare functions, although good ones don't really exist as I explained in my article on Arrow's Possibility theorem I mentioned here before I think).

For example, John Rawls famously argues for maximising the utility of the least well off member of society. So, we'd prefer a society with utilities 2,2,2,2 to one with utilities 1,10,10,10 because it's unfair for the one guy to get the bum deal. Similarly, we'd prefer a society with 2,2,2,1000 to one with 1,3,3,3. In other words, we tolerate even gross inequality if it betters the worst off as well as the best off.

OK, I could go on for hours about this and hundreds of related points. Perhaps if, as seems likely, a massive discussion grows out of this, we'll talk about them a bit.

By Philip Ellison on Sunday, January 06, 2002 - 10:48 am:

Are all Cambridge undergraduates socialist? (quick poll please!) Also, what exactly is a society that does not enforce equality but IS egalitarian? (as this sounds like good thinking to me, but I can't see how it would work... though I may just be getting confused over the precise political terminology).
Thanks

By Dan Goodman on Sunday, January 06, 2002 - 06:47 pm:

Hah! Short answer, no. In fact, very few Cambridge undergraduates are socialist. Most are apolitical and of the political ones socialists are in a minority.

An example of a society that does not enforce equality but is egalitarian is one that rewards according to the amount of effort put in. Although such a society would not be exactly equal (some would choose to put more or less effort in), it would be egalitarian since there is nothing to stop people putting more or less effort in.

By Arun Iyer on Sunday, January 06, 2002 - 06:51 pm:

mathematically speaking,
that is a beautiful perception of what a socialist economy aims at!!!

however,i remember a sentence which comes from no one else but BUDDHA.....
"Desire is the root cause of all sorrow"

there is no way that a man could stop desiring so...the result is______________!!(guesses anyone)

love arun

By Philip Ellison on Sunday, January 06, 2002 - 08:00 pm:

Does the Cambridge Socialist Club still exist, and if it does is it (or was it) anything to do with the uni? I ask because I've just read a book about Philby and friends, and there was some ambiquity as to whether the Club was to do with the uni (not necessarily officially, but perhaps as a collection of students). Any info. would be appreciated.

By Dan Goodman on Sunday, January 06, 2002 - 11:59 pm:

I've not heard of a Cambridge Socialist Club. There are lots of university societies that are broadly left wing though. You can find a list of the official Cambridge clubs and societies by going to the university main page http://www.cam.ac.uk and looking for it (not too difficult to find I think). Particularly active at the moment is CamSAW (Cambridge Students Against War), but there are plenty of others.

By Brad Rodgers on Monday, January 07, 2002 - 02:14 am:

First, I enjoyed your proof and your following commentary. I have to ask, has any sort of a mathematical argument for eglitarian values ever been presented before. I ask because I think that if one were, it would probably be popularized more...

Now to the commentary; I think that "your" first point, the one made by most capitalists, is probably the best, perhaps the only, way to argue against a mathematical presentation like this. I'm not entirely sure that such an argument would necessarily mean less happiness overall (I'll have to think about this, probably for a while, since school is starting back up), and I'm not even sure that human nature fundamentally dictates only working for ones own gain; though class and immediate reward is present in the written history of man, Georg Maurer proved that common ownership of goods and lands was the foundation and starting social organization of Teutonic peoples. Similar structures have been found to exist anywhere from Ireland to India.

I think, unfortuneately, the sustinance for such relations was a nationalistic militant enthusiasum. This nationalism, whether nation is considered to be those sharing class interests or those sharing militant interests, seems to be omnipresent, and we seem to trade one definition of nation for the other. Certainly, neither is good, but the only way to dissolve either is by ending class or ending military. Whether this is historically possible, I don't know. I think nationalism is an infantile disease, but nonetheless a disease not easily curable.

I think your idea of a society that is naturally eglitarian is the best way to achieve such a state, certainly not by replacing an monetary upper class with a ruling upper class, neither one being particularly intellectual.

And, to address your third point, I think it only makes sense to maximize utility, not necessarily to distribute utility equally. Fortuneatly, at least given a set wealth, these are the same :-)

Hope to continue,

Brad

By Dan Goodman on Monday, January 07, 2002 - 02:55 am:

I've not come across a mathematical argument for egalitarian values in the strict sense of the above, i.e. that equal distribution of wealth implies maximal total utility. I have to say I think it's likely that it has been done before though. By the way, what it does prove, if you accept the validity of your characterisation of the utility function, is prove that amongst all social systems capable of generating a total amount of wealth X, equal distribution of wealth produces the highest total utility. (You need one more dubious assertion actually, that everyone's utility function is the same, which is not necessarily true. A greedy money-lover might get more utility from additional wealth than a hippy for example, who may get no utility from additional wealth, maybe even negative utility!)

I agree that it is not a fundamental part of human nature to work only in your own gain. There are so many examples of this, Nrich is one. However, it can be an interesting exercise thinking about different social systems on the assumption that people will do this. They may behave better, but they won't, in general, behave worse.

On the third point, you might like Rawls' book (I think it's called "A Theory of Justice"). I have to confess I haven't yet read it, but I mean to. Also, I'm pretty sure that his condition is also the same (given your conditions on the utility functions of people) as the maximising total utility condition. Maximising the minimum utility will, I'm pretty sure, lead to all wealth (and hence utility) being equal if the utility functions are convex, increasing, same for everybody, and so on.

By William Astle on Monday, January 07, 2002 - 04:41 pm:



This is interesting.

Is nationalism necessary to sustain egaliteranism? I don't really know but I'd have thought most examples of egalitarianism occured in stateless societies where there was no need to enforce property rights etc. Native American cultures I think had very loose notions of property and ownership.

You've made another assumption that I think is dubious. You constructed a (sociatal) utility function G from the utility function of individuals f by summation. Dan pointed out that different people have different utility functions. Even if you assign a utility function fi to the ith individual and construct G by summation (why did you choose summation by the way?) or some other function g(f1, f2, ...), that for reasons of equity it is probably fair to assume is unchanged under permutations of the fis, so that G(a1, a2,...)=g(f1(a1), fa2(a2),...) we still have the problem that the utility of one individual is not necessarly independent of another's wealth. The hippy might be satisfied with a frugal life but the excessive wealth of the capitalist will probably offend him or her. It is a little ironic that this argument for equity of wealth can hold on the basis that each person has a complete disinterest in the happiness of others.

Even if we can find a wealth deistribution that maximises a sociatal utility function G it would still be a problem to find G itself. I am pretty economically and politically ignorant but as I understand it one of the main arguments used by those in favour of the market system is that it allocates resources in a way that maximises the overall capital return (over some period). The utility function is automatically found. The problem then I suppose is whether it is the correct utility function to maximise.

By Philip Ellison on Monday, January 07, 2002 - 05:01 pm:

It's not a fundamental part of human nature to work solely for your own monetary gain, though with a broad enough description of gain, nearly everything you do could be interpreted as being for your own gain in some respect. The difference just comes in what sort of "gain" you are after (and whether it is at the expense of others, I suppose).

By Arun Iyer on Monday, January 07, 2002 - 07:12 pm:

In the present view of world,
we cannot study the above prescribed situation just in reference to the human nature.......situation is the most vital player here....situation,the environment compels you to change yourselves whosoever you are....

Now a socialist economy,proposes to achieve what BRAD has written through a mathematical statement and its been well proved by DAN....So should we assume that socialist economy is the best as regard to the capitalist economy or the mixed economy?????

the question is quite hard to answer...
here's where human nature and various situations come into play....Let us have a look at them...
1.Some people prefer socialist economy...Such people usually have the tendency to share things and aren't bothered by the thoughts of distrust i.e such people are very trustworthy and expect others to be so....These people have such a nature because of the environment in which they are brought about....(i will let people here analyse the situation)

2.Some people prefer Mixed Economy.....such people don't share things easily and neither are ready helpers..infact they demand some sort of proof of trust before they go ahead and help..
Again there are situations which make these people what they are....Usually you will find such people in developing countries like India...

3.Some people prefer capitalist economy....such people are quite straightforward.... whatever these people do is in form of a business....in other words these people are very business minded people...

i can go on and on about these things infact many things are to be discussed which are very much the cause of what the world is today!!!

From here we can actually shift ourselves to psychology as well wherein we can study the exact environment in which various people live!!

love arun
P.S->If you can visit India and have sometime to ride in one of our local trains,then you can study various facets of human life.....Life in India is quite interesting in one way to speak!!

By Brad Rodgers on Wednesday, January 09, 2002 - 12:45 am:

When I use the term nation, I use it in a very loose sense, taking it to mean 'putting precedence of one social organization/body over all others'. So in this sense, native americans were nationalistic; not in the contemporary sense, but in the sense that they had tribal "wars", and sought aquisition of something (not necessarily land) from another group. A similar sort of 'nationalism' applies to the capitalist class today. They care little about those with less wealth than them, however apparently try to act in a way to preserve their "equal's" wealth. This is plainly seen in political decisions in a representative government.

I think to some extent this social organization into small groups, or at least not all-encompassing groups, is unavoidable. It clearly possesses evolutionary benefits, and it probably is why the social animal is able to survive better than the non-social. That is to say, I believe 'nationalism' in a non-national sense is innate.

The reason I chose to sum utility functions is because utility is almost certainly additive. It simply makes no sense to me otherwise...

I would expect that most people would have similar "happiness-from-money functions", although, I really have nothing I suppose to support this.

Brad

By Brad Rodgers on Friday, January 11, 2002 - 02:34 am:

I didn't notice this before, but this theorem is actually a more general version of the AM-GM inequality. We know ln(x) is concave down, therefore, if a1+a2+a3+...+an=1, then

ln(a1) + ln(a2) + ... + ln(an) £ n ln(1/n),

or condensing both sides then raising to e,

a1a2a3...an £ (1/n)n = [(a1 + a2 + ... + an)/n]n

Upon multiplying by cn, and setting axc=bx, which provides for any distribution of bn,

(b1b2b3...bn)1/n £ (b1 + b2 + b3 + ... + bn)/n

Interesting eh? I'm looking for other non-trivial results, but this is nice because it's the only proof of the AM-GM I've seen with an intuitional theorem at the roots.

Brad

By Brad Rodgers on Friday, January 11, 2002 - 02:47 am:

Actually, searching through Mathworld, I've realized that what we've been using is known as Jensen's Inequality. It was only discovered in 1906, and I have no idea how it's normally proven, unfortunately. Interesting that it would have applications in economics...

Brad

By David Loeffler on Saturday, January 12, 2002 - 01:11 pm:

Normal proof is by a somewhat nasty induction method. In fact, you can take just about any of the standard proofs of AM-GM and adapt them to Jensen's inequality; my favourite is Cauchy's "backward induction" (prove the theorem for n = 2m, then show that if it holds for n it holds for n-1).

David

By David Loeffler on Saturday, January 12, 2002 - 01:14 pm:

(This "backward induction" is what Hardy, Littlewood and Polya use in their book "Inequalities" - it's the standard work in this field. Note that this is "inequalities" in the mathematical, not the sociological, sense!)