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# Why Income Mobility is Larger in the Middle

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Whenever we talk about income mobility, we should never forget that we're talking about mobility in each direction. One can move down the income scale as well as up. Indeed, if we measure income mobility of households by movement from one quintile to another, it necessarily must be the case (given a fixed number of households) that for every household that moves up, one must move down. If that didn't happen, then the apparent quintile into which more households moved than moved out of would not really be a quintile.

Because movement can be down as well up, income mobility for the bottom quintile will likely be less than for the middle three quintiles. Why? Because a household in the bottom quintile cannot move to a lower quintile. It can only stay in the same quintile or move up.

Similarly, income mobility in the top quintile is likely to be less than for the middle three quintiles. The reason: a household at the top cannot move up but can only stay in the same quintile or move down.

Here's how Alan Reynolds put it in his excellent blog post, "Likely Sources of Obama's Misconceptions about Income Mobility":

The middle three quintiles are defined by both a floor and ceiling. Unlike the middle quintiles, however, movement in or out of the top or bottom income groups can be in only one direction. Anyone in a top income group in any particular year must have either been in the same or a lower group in previous years, because there is no higher group to move down from. Anyone in a bottom income group must likewise have either been in the same or higher groups in previous years, because there no lower group to move up from.

This simple mathematical distinction has led many careless observers to deplore the illusory fact that there appears to be less "mobility" among rich and poor than there is in the middle. Rather than indicating that the poor are stuck at the bottom and the rich secure at the top, this is simply the unavoidable consequence of the fact that only families at the top and bottom can move in only one direction. Like so much overheated rhetoric about inequality and mobility, this is just another example of people forming extremely strong opinions on the basis of extremely weak logic and evidence.

CATEGORIES: Income Distribution

Ted Levy writes:

In a sense, it's even worse (or more distorting) at the top. After all, there is a specific amount of earning those in the bottom quintile need to achieve in order to move into a higher quintile. But there is NO amount of additional money you can earn in the top quintile to push you into a higher quintile. Bill Gates could double his money and not move out of his quintile. So the putative stability of this quintile hides the fact it is open-ended at the top. The apparent stability is related to the fact we are not looking at sextiles.

ed writes:

This would be more convincing if you had some statistical model of the income process, or class of statistical models, for which your claim holds.

It is simply not true that your claim must hold for all statistical models. Trivially, if income in each period is i.i.d. (independent and identically distributed), then the mobility out of each quintile would be the same, despite the fact that "families at the top and bottom can move in only one direction."

Jody writes:
This would be more convincing if you had some statistical model of the income process, or class of statistical models, for which your claim holds.

Random walks.
http://en.wikipedia.org/wiki/Random_walk

Basically, assume memory.

Dylan writes:

Really good post, David. People, and intellectuals particularly, get so fixated into the small letter, minor issues, abundant data and irrelevant facts, that they tend to forget the most basic self-evident realities. It is a good example of scientism, if you ask me.

Fidel writes:
Indeed, if we measure income mobility of households by movement from one quintile to another, it necessarily must be the case (given a fixed number of households) that for every household that moves up, one must move down. If that didn't happen, then the apparent quintile into which more households moved than moved out of would not really be a quintile.
I think this is not necessarily true. Imagine a situation where:

- One household moves from q1 to q2.
- One household moves from q2 to q3.
- One household moves from q3 to q4.
- One household moves from q4 to q5.
- One household moves from q5 to q1.

The number of households is fixed, but there are four upward movements and only one movement downwards.

ed writes:
Random walks.
Sure, this property holds if income is a random walk, but I do not believe income is a random walk. (And if it were a random walk, that would probably be a reason to be much MORE concerned about inequality!)
Basically, assume memory.

I do not believe this is not sufficient to guarantee the property described, for any reasonable definition of "memory."

To be clear, I think the idea here is basically correct, but it is being stated in a too-strong and too-sloppy way.

writes:

Wait, what? If the model inherently requires that the top 20% can only move down, that seems like a pretty serious, fixable limitation!

Doesn't that call for either:

a) Allow infinite movement in either direction (real income on a logarithmic scale)

b) higher granularity so that you get unidirectionality for a smaller portion of the population.

I find it hard to believe that this kind of thing is distorting all the reports of income mobility.

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