Economics: Utility and Wealth Redistribution

In my last post, I created and discussed a definition of wealth redistribution. If you haven't read it, at least take a look at the definition given, as it will be important for this discussion.

Utility

Utility is an economic concept that attempts to quantify the value of things. Unlike a more ordinary quantification attempt, however, it is merely ordinal, and not cardinal - that is, utility does not come in discrete meaningful units.

Therefore, the real purpose of utility is to determine which in a group of things is the most preferred option. While this might give us an idea as to the degree of difference in preference between the various choices or objects in our pool, it does not tell us the absolute difference in utility between them.

I'm sure that seems quite limiting to some of you. For the purposes of our current discussion, however, it does more than enough, so we shall forge ahead.

Utility and Wealth Redistribution

So, what are the different utilities and changes in utility present when wealth redistribution occurs?

Well, for the moment let us assume we are looking at a single act of wealth redistribution between two individuals - one with one million dollars, the other with no money at all. No other parties are involved, and we will ignore all impact to other parties as a result of the redistribution - what are normally called externalities in economics.

What sorts of utility-related things do we have in this situation?

Well, each of the two involved parties is sitting at a particular utility as a result of their wealth. It should be fairly obvious that the person with one million dollars has a higher utility than the person with zero dollars.

How much higher? We don't know. If we had additional data, we could plot the utility of both parties at every possible amount of wealth between zero and one million dollars, and we could then create some semblance of a cardinal system, if only because we could compare so many discrete ordinal values.

There are a number of important theories about utility we could cover here to make the work simpler. I'm going to neglect that, however, in favor of a different approach - instead of examining the utility of each party at many arbitrary values of wealth, we will use the utility of a change in wealth.


This approach is better for several reasons. Perhaps most importantly, it more accurately represents the change in utility that occurs when wealth is redistributing. Outside of that, it also encapsulates a very important concept when it comes to utility - that the possibility of losing wealth carries with it a greater loss in utility than the positive utility associated with gaining wealth.

I will not be covering the bones of this topic myself, so you can either take it on faith or read this paper, which, coincidentally, also covers both the theory of marginal utility and the idea of isolated decisions.

By combining loss aversion and diminishing marginal utility, we can come to some valid overall assumptions about the utilities associated with gaining and losing wealth.

For the poor man, who has no money at all, gaining money will cause him to gain utility, with the amount gained generally diminishing as his wealth increases. His curve for a change in wealth would look something like this...

The rich man loses utility as he loses money. According to loss aversion he will value equivalent losses at a greater magnitude than equivalent gains. However, diminishing marginal utility also means that any loss or gain should be generally less in magnitude than similar losses or gains by the poor man, at least until the rich man is losing a great deal of money. A graph for changes in wealth for the rich man might look like this...

Equilibrium

Now we have some stuff. What do we do with it?

Well, what we are concerned with is the point of optimal wealth redistribution. This can be defined - in this limited instance, at least - as the amount of wealth transferred from the rich man to the poor man that leads to the greatest total utility between the two of them.

Unfortunately, the problem is not trivial, even in this limited case - trivial meant in the literal mathematical sense of requiring no work to find our answer.

This is because a redistribution of wealth from one party to another does not always change net utility the same way.

For instance, we know that taking a small amount of money from the rich man and giving it to the poor man will cause an overall increase in utility - although it is difficult to know how much. On the other hand, because of loss aversion and diminishing marginal utility, there will eventually be a point at which you are taking away too much money from the rich man and the utility gained by the poor man cannot make up for it.

So, how do we solve the problem?

The graphs given are those of the change in utility of wealth from one point to another. To find a local maximum or minimum, you simply need to find a point at which the values of both graphs sum to zero. Under ideal circumstances, only one such point should exist - if there were more, you would simply have to find every possible value and see which resulted in the highest total utility.

That or just use a search algorithm and add random restart to it.

So there you have it, right? The solution to socially optimal wealth redistribution is just that easy!

Other Concerns

Of course, the problem isn't that simple. Here are four reasons, picked randomly from a hat...

One, we are ignoring all externalities. This includes thins such as a possible negative social backlash from the wealth redistribution, a loss of job creation, a negative impact on industries that produce luxury goods, and a great deal of other possibilities. Perhaps most importantly, how others feel about the redistribution of wealth - and its degree - can be a major positive or negative externality.

Two, we are ignoring the possibility of more than two individuals. If, for instance, we were transferring money from one rich person to two poor people, our solution would be very, very different. (To be exact, both poor people would end up with slightly less money, but more overall wealth would be redistributed - at least under the assumptions in our simple simulation.) You can imagine how dealing with an entire nation with a great many rich and poor people would be difficult - especially when you take into account the fact that they are all rich and poor to different degrees, and have their own personal utility curves.

Third, we are ignoring the possibility of multiple transfers which, because of how most people isolate decisions, could have a vast positive or negative impact on our equilibrium - and would certainly change it in some way. For instance, if we treated our wealth in the above example as income, and the transfer as a yearly transfer, how would our equilibrium change?

Fourth, we may be looking at the utility of wealth in entirely the wrong fashion. For instance, many studies have shown that utility is relative, rather than absolute, when you are looking at wealth. That is, the wealthiest members of society are always happier than the poorer ones, simply because they happen to be better off in comparison. There are a number of other alternative views of utility in relation to wealth, and all of them cause us to have different assumptions and, hence, different results.

And, of course, there's another very large problem - how do you get the graphs we had above to begin with? You see, everyone has their own personal utility curve when it comes to wealth - or anything, for that matter. That would mean we'd have to find a utility curve for every person in America if we wanted to solve the problem for the entire nation, and that just seems downright impossible. Right?

Luckily, it's not actually impossible. Aggregate utility curves can be created through the use of surveys and some fancy econometric techniques, most of which were developed for valuing non-market goods related to the field of environmental economics. Unfortunately, creating such a survey and administering it properly is not something I have the resources to perform on my own, so that idea will have to be scrapped for sometime after I write this all up and get a huge grant to pursue my interests in quantitative welfare economics.

Cause that'll totally happen.

Summary, Continuation, and Assistance

On the one hand, this post describes the idea of optimal wealth redistribution in a scenario with only two individuals, and provides a solution. On the other hand, that scenario is entirely inaccurate and mostly useful only as a demonstration, and the best way forward is not one that can be taken as of the moment.

Moving forward, then, I'm going to do two things.

First, I'm going to write up an aside about a theoretical economy in which all income is redistributed evenly and all ventures are crowd-funded. It's interesting and, oddly enough, relevant to the current discussion. Why? Because one possible externality that exists from wealth redistribution is the potential drop in job creation - or, at least, it's an argument you hear quite commonly. This examination should help to determine if it is valid or not - as of the moment, I'm not sure on way or the other.

Second, I'm going to attempt a very rough estimate of socially optimal wealth redistribution. This will be done  in a fancy 'too much or too little' approach, whereby I will attempt to determine if certain areas would prefer more or less wealth redistribution.

So, what sorts of assistance could I use from the handful of people actually interested in this? If you know of any good places to get statistics for things such as local/state/federal taxation in western developed nations, amounts given to charity by residents of a certain area, rich/poor gap, or anything else you might think is relevant, I'd love it if you could let me know, because one of my primary goals right now is finding good numbers for a lot of these statistics.

Anyways. As always, I hope you enjoyed reading this, and maybe learned something. Here's your pun of the day, and hopefully I won't be a sieve and write for four days just to get the next post out...

Pun of the Day: I really enjoy conversations with nervous people. Whenever they go 'But, um...' before saying something, I say 'ch!' and smile while they look confused. It's great.

Not quite a pun, but I still find it quite funny. Or maybe just fun. Not quite sure on that one.