## Thursday, September 08, 2011

### On inequality of wealth

Note: Non-sports post.

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In the United States, the top 10% of the population earns 30% of the income. And the top 10% of the population owns 70% of the wealth.

Statistics like these seem to be popping up all over the place lately ... someone I know posted one on Facebook a few days ago, and there was a newspaper article or two in the last month. I'm not sure what happened to bring all this up. (If anyone has links from the last week or two, let me know. I can't find them at the moment.)

A couple of years ago, taking about the Gini Coefficient, I made a bunch of arguments about why the distribution of income doesn't matter much. I think it matters a bit, but not much.

Here, I'm going to concentrate on the distribution of *wealth*. For wealth, I'm going to argue that, given a particular distribution of income, the distribution of wealth is almost completely meaningless as a moral issue, or an issue of people's well being. That is: criticize, if you want, the fact that the top 10% get 30% of the income. But given that income distribution, *it doesn't matter* how much of the wealth the top 10% own: whether it's 10%, 30%, 70%, or 99%.

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The difference between income and wealth is that income is a rate, how much you earn in a particular year. Wealth is the total amount that you possess at a specific time.

How does anyone gain wealth? Other than inheritance (which we'll disregard here), you have to save or invest some of your income. You can earn ten million dollars one year, but if you blow it all on cocaine and hookers, your wealth will be zero.

So your wealth is a result of three things: (1) your income, (2) the amount you save, and (3) rate of return on the amount you save. As I said, if you hold (1) as fixed, wealth is affected by only (2) and (3).

Suppose you have two people, John and Mary. They have exactly the same education, and they graduate into exactly the same job, paying \$50,000 a year. John spends all his money every year. Mary saves an annual \$6,000 in a retirement fund, earning 5%, and spends the rest.

What happens? After 40 years, John has \$0 in wealth. Mary has \$725,000.

Is it fair to complain about that? I don't think so. Sure, Mary is now (fairly) wealthy while John has to live on just Social Security. But, in the past, John lived much better than Mary, to the tune of \$240,000 -- \$6,000 a year for 40 years. Some people's tendency would be to take some of Mary's money and give it to John. But that wouldn't be fair. It would actually be quite an injustice. Mary deliberately lived significantly worse than John for 40 years, just so she could have a better retirement. Giving that money to John would *compound* the inequality, wouldn't it? It would take from the (formerly) poor lifestyle and give to the (formerly) rich lifestyle. It would compensate for the future where Mary spends more than John, but not compensate for the past, when John spent more than Mary.

Really, even though Mary has more money than John, over their lifetimes, they're equal. Thirty-five years ago, John spent \$4,000 on a new state-of-the-art TV. He knew, when he bought the TV, that \$4,000 then would be the equivalent of \$22,000 at retirement. He bought the TV anyway. Nothing wrong with that. He chose, freely, to live \$4,000 richer than Mary back then, in exchange for living \$22,000 poorer than Mary later. Mary also knew the terms of the trade, and made the other choice.

But, over their lifetime, they are exactly equal. \$4,000 can buy a lot of things: a vacation, a TV, a boat, a motorcycle, or a retirement fund of \$22,000. If John had bought a TV, and Mary had bought a boat, could anyone argue that Mary is richer than John because she has a boat? Of course not -- because, by the same token, John has a TV of equal value.

The same thing applies here: if John has a TV that costs \$4,000, and Mary has a \$22,000 retirement fund, which also costs \$4,000 ... they must be equally rich, right?

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When you talk about the distribution of wealth, what you're really talking about, for the most part, is the distribution of a desire to save. And there is no "proper" distribution for that, any more than there's a "proper" distribution of religious beliefs. People are diverse, and they have different tendencies. Some people like to spend, and some people are compulsive savers. Humans choose differently from each other.

Suppose the 1% of the population that owns the most rare baseball cards happens to own 70% of the rare baseball cards. It just means that the other 99% don't care as much about baseball cards. If they have the same income, they just own more other things instead.

Now, you can still argue that the reason it's a problem that the top 10% has 70% of the wealth is that the bottom 90% doesn't earn enough money to be able to save. But that argument is better made by arguing about the *income* distribution. Because, otherwise, you're combining two issues: having money, and choosing to save it. If you were to complain that the top 1% own 70% of the baseball cards *because they have a higher income*, you'd be mostly wrong. Yes, the top 1% of baseball card owners probably DO have a higher income. But that's not the main reason they own 70% of the baseball cards. The *main* reason they own 70% of the baseball cards is because they really, really like baseball cards.

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Here's a model, for a numerical example.

Start by assuming a population of 10,000 people. They all have exactly the same education, and they all graduate at age 25 into a job that pays \$40,000 a year. They work until they're 65, at which point we measure their wealth.

But they're not all the same, because they have different personalities, and characteristics, and desires. Specifically:

1. They vary in how much money they like to spend. The mean of the population is to spend 90% of their salary and save 10%, but with a standard deviation of 15 points. Nobody saves more than 50% of their salary, or spends more than 115%.

2. They vary in how many children they want, and when. 20% of them want no kids. 20% of them want one kid early in life (age 27), and 20% want one kid later in life (age 35). 20% of them want two kids early, and 20% want two kids late. Kids cost \$5,000 a year in expenses to age 18, and then \$20,000 annually for the next four years, all of which comes out of saving.

3. They vary in how good they are at investing their money. Some play it safe, and some are more aggressive. Some study investing, and some don't. The average annual return is 4%, with an SD of 1.5 percentage points.

4. They vary in how much effort they put into their job, which affects their annual salary increases. The mean increase is 2% a year, with a standard deviation of 0.5%. Nobody ever gets fired or earns less than \$40,000.

5. Nobody goes into debt more than \$50,000. Once they reach \$50,000, they cut their spending to keep the debt at \$50K. All debt is paid off in the year before retirement. Debt earns interest of 10%.

Under these conditions, I ran a random simulation of the 10,000 people.

So, at age 65, what percentage of total wealth will the top 10% own? Take a guess before reading on. I'll write it cryptically so you don't see it by accident when you're thinking.

The top 10% of these graduates own (7 * 9 - 22)% of the total wealth.

Got that? It's not as big as the real-life answer of 70%, but it's pretty big nonetheless. And it's *completely* due to the decisions of the individuals themselves. There is no inequality, no racism, no bad schools, no corruption, no government favors, no explotation by greedy employers. It's just natural variation in how human beings choose to live their lives.

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Some of the other results:

The top 1% had 7% of the wealth.
The top 10% had 41% of the wealth.
The top 50% had 99% of the wealth.

As you would expect, the wealthiest people were the ones who saved the most and got the highest rates of return. The wealthiest, person number 7,490, wound up with just over \$4,000,000 in wealth. She saved 41% of her salary and earned 8.3% per annum. In case you think 41% is a lot ... it's not, really. There are a lot of misers in the world. At retirement, this person earned \$55,551, which means she was living on around \$33,000 per year. That's not unreasonable for an outlier, just over 2 SD from the mean.

Overall, it turned out that number and timing of children didn't matter much. Neither did salary (although the salaries were all pretty close). Some of the richest people earned below-average salary increases.

So what mattered is how much they saved, and how well they invested it. Of course, my model is way oversimplified, but that does correspond to my perception of how wealth happens in real life, where my sample of friends earns around the same as I do.

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One thing I should note is how the "top 1%" figure of 7% is way, way off the real life figure of 38%. Why is that? Well, the main reason is probably that the model didn't consider the possibility of enterpreneurs who can occasionally create a multi-billion-dollar company out of nothing. If Bill Gates and Warren Buffett were in the model, the figure would jump substantially from 7%.

What's more surprising, I think, is that the top 10% number was so high, at 41%. I expected it to be much lower, considering that there's so much less variation here than in the real model:

1. Here, everyone had roughly the same income, between \$40,000 and \$60,000. Real life, on the other hand, includes sports stars, CEOs, and other people with high productivity.

2. Here, everyone was 65. Older people are obviously wealthier, since they've had much more time to earn and save. If you take these 10,000 retired people, and combine them with 10,000 babies, then the distribution is much more unequal, since you've added a bunch of zeroes. Then, the top 10% jump to from 41% of the wealth to 64%.

The age thing is a big issue. Even if everyone were exactly equal in every way, following the same career path and the same wealth accumulation path, the distribution would be unequal if you take a snapshot in time. You'd be combining 65 year olds who are rich because they've been saving, to 25 year olds who WILL be just as rich, but aren't yet. (That, by the way, is why it's best to look at lifetime income, or at least age-adjusted income, instead of snapshot income or snapshot wealth.)

Oh, and by the way ... I built a certain amount of progressive taxation into the model. I assumed all salary above \$40K is taxed at 30%. I also assumed that the savings rate is based on after-tax salary. And finally, I assumed that if you save more than 30% of your after-tax salary, any excess is taxed *again* at 30%. (This was easier than trying to compute tax on investment income.) The numbers above are *after* all this progressive taxation.

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So, my argument boils down to something like this (directed at a random skeptic):

You say that the top 10% owns 70% of the wealth, and that's too much. Why is that too much? It can't be just inequality, because here I have a model where everyone is equal, and the top 10% still owns more than 41%. Why do you think 70% is wrong, and what should the number be, and what are your assumptions?

And suppose I cornered you, and asked you to tell me exactly what your policy prescriptions are -- how much to tax the rich, what to do with the money, how to tax investment income, what loopholes to close, how to get the poor to save more, and so on. Then I would ask you, "after all that, how much of the wealth would the top 1% own?"

I'd bet you couldn't answer that. And if you don't know what the distribution of wealth would be in your ideal world, how can you possibly argue that it's the wrong number now?

At Friday, September 09, 2011 12:27:00 PM,  mettle said...

A few issues:
First, you state that it doesn't matter how wealth is distributed. You then go on to show that you get a skewed distribution based simply on savings rate. That does not show that wealth distribution doesn't matter - it shows something completely different.
Second, 40% and 70% are vastly different. Another way to look at it is that the bottom ninety percent will have 60% or 30% of the wealth. The former seems reasonable; the latter ridiculous. Indeed your discussion is a great way to make the argument that the number should be at 40% - again, vastly different than 70%.
Finally, and most damningly, one of the big issues you ignore is wealth transfer across generations. What you need to do is to do your simulation with multiple generations. So first, you'd see a further concentration of wealth - after a few generations you might get the top 10% owning 90% of the wealth and unless I'm missing something, this concentration would continue unboundedly. Are you really arguing that because you get it in a simulation, that it's not a problem?
The obvious policy prescription one could prescribe is that inheritance tax should be, say, 80% of net worth above \$.5mil. You could determine what rate you need to keep things at 40% or at least stable.
So, I think you've really shown the opposite of what you intended.

At Friday, September 09, 2011 2:05:00 PM,  Dave said...

I think the problem is inequality at the outset. If, like your models, we all started off from the same point on education and salary there wouldn't be a reason to be mad about who chose to spend and who chose to save. But this is not what happens in reality. The already wealthy tend to get the better education and jobs. They make more income and are able to both spend and save more. Rinse and repeat and you have the inequality in income and wealth we have right now.

The people who don't like that should be looking for ways to make it easier for the non-wealthy to get the better education and thus better jobs. It would also help if those without great incomes could save more. But our society is so conditioned toward having to buy really expensive things like houses and cars that its really difficult for anyone to save and accumulate wealth. If we had cheaper houses and cars I think it would be easier to send your kid to a really good college and get the ball rolling on becoming one of those wealthy families that starts off ahead of the game and just keeps adding to their lead.

At Friday, September 09, 2011 3:15:00 PM,  Zach said...

In addition to the dynastic wealth issues, income from wealth is taxed at a lower rate than income from work.

Both dividend and LTCG income is taxed at a preferential rate to ordinary Income.

This tax structure further ensconces wealth. The wealthy get wealthier based on nothing more than being wealthy.

At Friday, September 09, 2011 11:46:00 PM,  Anonymous said...

The first comment by mettle is really all that needs to be said. If reality matched your model's distribution we wouldn't see inequality statistics popping up all over the place.

When you talk about the distribution of wealth, what you're really talking about, for the most part, is the distribution of a desire to save. And there is no "proper" distribution for that, any more than there's a "proper" distribution of religious beliefs.

So, you assume the proportion of salary people spend is always rational and simply reveals their preference for present vs. future wealth?

3. They vary in how good they are at investing their money. Some play it safe, and some are more aggressive. Some study investing, and some don't.

Do you believe variation in investment income for people in a given income range reflect some actual skill??

You say that the top 10% owns 70% of the wealth, and that's too much. Why is that too much?

The poor get greater utility from that wealth than do the wealthy.

And if you don't know what the distribution of wealth would be in your ideal world, how can you possibly argue that it's the wrong number now?

This is a non sequitur, but I'll take a crack anyway: historical context. Inequality has been growing. Is it your contention that the John Galt producer class has grown recently? Or merely that they're more elite than they were 40 years ago?

At Monday, September 12, 2011 10:47:00 AM,  notjaybilas said...

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At Monday, September 12, 2011 10:49:00 AM,  notjaybilas said...

That your models assume that everyone was born on the same (first, second?) base is a pretty strong signal that your parents hit for extras.

At Monday, September 12, 2011 9:21:00 PM,  Aporia said...

"3. They vary in how good they are at investing their money. Some play it safe, and some are more aggressive. Some study investing, and some don't. The average annual return is 4%, with an SD of 1.5 percentage points."

I refuse to believe that an ordinary person, even if they are merely two standard deviations above the mean in investing ability (but not professional investors analyzing stocks for a long/short hedge fund or Warren Buffet), can consistently extract 3% alpha, which alpha is defined as returns associated with the skills of the investor, (not benefiting from "beta" by merely investing in the general stock market during a rising stock market/ or being in bonds during an economic slowdown) from the market. I don't think mutual funds can provide such an advantage to ordinary investors because their ponderous size precludes them from allocating assets without affecting the price of the asset they purchase or sell. Furthermore, they are also competing against other mutual fund managers with similar information. (Macro funds, in contrast, can take relatively large position in the currency, index futures, and bond markets because those markets are relatively liquid unlike the market for most individual stocks). Also, those who are invested in riskier assets such as stocks may have a higher return in the long run (this extra return is not associated with alpha) because they benefit from the risk premium inherent in holding more volatile assets. Let's say that the equity risk premium is 3.5% (this number varies historically and regionally); this would mean that difference in expected return between a person holding a portfolio of 100% government bonds and another person holding 100% equities (assuming this person is trying to profit off of beta and has not ability to extract alpha) is 3.5%, meaning the difference in expected returns among the 1%tile and 99%tile (a gap greater than 4 SD) is only 3.5%, lower than what would does your model would predict. In essence, I am invoking a form of the efficient market hypothesis to assail the notion that an ordinary person can consistently earn a return 5% above normal every year.

In the real world, there is substantial covariance in savings rate and income, but your model does not need to make that into account because it purposely reduced the variance in some (it assumes an equal starting income and a low SD for the rate of income increases). In fact the six-sigma ratio of salaries after 40 years for a person who has a salary increase of 3.5% a year and 0.5% a year is 3.2 (1.035^40/1.005^40 ; I assumed yearly not exponential compounding); moreover, as you mentioned, income above \$40,000 is taxed 30%, so a the 3 SD person has more of his/her income taxed than 3 SD person, furthering reducing income inequality.

At Thursday, September 29, 2011 11:25:00 PM,  Paul Thomas said...

Your model appears to show that a system which corrects for most* unearned advantages produces far less concentration/inequality of wealth than one which does not.

That would seem to imply that correcting for those advantages is a good thing, n'est ce pas?

*I say most, because inherent propensity to save resources is a strategy which is favored in modern life to a degree which was absent when humans evolved. People, as a whole, don't behave like rational actors when it comes to wealth. They spend it like drunken sailors. The human brain inherently overvalues \$4000 today (utility-wise, I mean) when compared to \$22,000 at retirement, which is why forced-saving and forced-insurance schemes like Social Security actually make sense as policy.

At Sunday, November 27, 2011 2:11:00 PM,  Kyle said...

The inequality of wealth is a never ending issue. It didn't affect much but at least we know the status of income distribution.