Monday, April 25, 2011

Wealth distribution: a model

The above chart shows the United States wealth distribution, alongside with what people think the wealth distribution is, and what people think it should be.  It comes from a paper by Norton and Ariely earlier this year, and it's made the rounds in the news.

That's pretty interesting, but I always find myself wondering how to interpret this.  Is the wealth distribution unfair, or are people just clueless about wealth distribution?  Probably both.  I feel pretty clueless about it myself.

To be honest, the very way the data is presented seems unintuitive to me.  I mean, I'm a physicist, so what I consider "intuitive" is all out of whack.  But seriously, the wealth owned by the top 20%?  I have very little sense of what to expect from such a number.  It has to be somewhere between 20% and 100%.  84% seems like too much, but what do I know?

If I were to pick a wealth distribution, I would start with a model, and from there calculate the percentage owned by the wealthiest 20%.  So here's my model: wealth has a normal distribution on a log scale.*  People who are one standard deviation above the mean own N times more than the median.  People who are one standard deviation below own N times less than the median.  N is a number that we can choose.  For graphing purposes, I am choosing N=2.718 (Euler's number).

In the above graph, the median wealth is 1, meaning that half of people earn more than that and half earn less.  But the mean wealth is actually more than that, since the distribution is skewed.  And the most common amount of wealth (called the "mode") is about 40% of the median.

The next step is to translate this to the amount of wealth owned by the top 20%.  So that's what I did for a few values of N:

(Note that though my plot has similar colors to the one at top, they aren't exactly the same since I split the top 20% into three groups.)

I showed N=6.5 because that's the number that seems to correspond to reality, according to Norton & Ariely.  I showed N=2.718, because that's the number I would have guessed if I had never seen the real data.  I showed N=1.5, because that's what strikes me as a "fair" distribution.  In other words, I would think it was fair if people who are one standard deviation above the median own 50% more wealth.  But in reality, they earn more than six times as much wealth.

I was surprised how similar my bar graph is to the one in Norton & Ariely.  I'm quite sure that most people answering this poll have no understanding of normal distributions or log scales, and I was all set to conclude that people are clueless.  I'm surprised to find that I agree with the popular opinion, because I think N=1.5 seems ideal.

Of course, the "ideal" value of N is completely arbitrary.  What do you think is an ideal value of N?  If you like, I can calculate the resulting wealth distribution.

*Some technical details: A normal distribution means that the probability density is exp[-y2/(2*log(N)2)].  But here, y is not the amount of wealth, but the log of the wealth.  When I transform from a log scale to a linear scale, the probability density becomes exp[-log(x)2/(2*log(N)2)]*1/x, where x is the ratio of the wealth to the median wealth.  This is the function I have plotted in the graph.  This is, by the way, just about the simplest model imaginable.


Larry Hamelin said...

Ah, you're getting interested in economics. :) And you thought physics was complicated! Come to the dark side, Luke!

There is no a priori "fair" distribution of income. We want to consider the effects of various distributions, especially the marginal propensity to save, and the marginal propensity to invest. And we want to compare that distribution to how much we want to save and invest.

miller said...

Larry, you have a much better understanding of this than I do.

Basic question: Does a higher value of N result in more or less propensity to invest (or am I misunderstanding your comment entirely)?

Larry Hamelin said...

A higher income generally leads to a higher propensity to invest, because of the decreasing marginal utility of consumption. Basically, the more money you have, the less the personal utility you can obtain from each additional (i.e. marginal) dollar of income.

Economics and evolution have a similar problem. It's very useful at a conceptual level to discuss economic concepts teleologically, as if there really were such a thing as "society" that actually wanted thing. But, like evolution, economics is talking about complex emergent behavior that isn't really teleological.

So we can say that if "society" "wants" to have a very high investment level, then we "should" have a very unequal distribution of income. Contrawise, if "society" "thinks" there's too much investment, we want a more equal distribution, so more is consumed than invested.

(FYI, as far as I know, N usually denotes the quantity of labor (i.e. the number of workers a firm hires, or the number of labor hours worked in macro) as one of the three factors of production. W usually denotes the wage rate, the dollar amount paid to labor. Cost of labor is thus W * N. The other two factors are L for land and K for capital (equipment, factories, etc.))

miller said...

I see. Does that mean that in your view, the poll results of what distribution people "want" are meaningless? I suspect most people answering the poll aren't thinking of it in terms of wanting more savings vs investments.

Larry Hamelin said...

Not meaningless, just metaphorical.

We cannot, for example, say that society as a whole really wants what the majority of people want, except as a metaphor. The reality is just that a majority of people want some particular thing.

Andy McB said...

I think the more important thing to consider when figuring out what seems fair is to look at the mode (most common value for income, wealth in-the-bank, etc). In a fair system, I would expect the average (median) guy to make about as much as is typical (mode). that even possible?

If not, I'd at least want the median person to be at the inflection point.

miller said...

I'm not sure how meaningful the mode is, since its location depends on whether we're considering a linear or log scale. I posited a distribution that is normal on the log scale, so in a linear scale the mode is somewhat lower than the median.

Larry Hamelin said...

Generally, economists don't look at income distributions in a way that locates or highlights the mode. The most common form of income/wealth distribution has percentage of people on the x-axis, and the cumulative percentage on the y-axis. An absolute equality of income/wealth would be represented as a straight line from (0,0) to (100,100) with slope 1.

The other common way of looking at income/wealth is to break it up by quintiles. If we look at income/wealth per person, the mode will obviously be at the top quintile.

Since an unequal distribution will by definition be right-skewed, the most common simple statistical measure is the median rather than the mean.

Ask me about this again in a year or so, after I've taken an econometrics course.