I apologize in advance if what follows is written badly, but the topic is touchy and I have to write defensively.
Assuming IQ measures what we call intelligence (which is a big assumption on which I have no particular position), it would stand to reason that high-IQ people would move up the income bracket scale at a higher rate than they would move down and vice-versa.
If that is true, then the eigenvector of the transition matrix would skew towards the top bracket.
I am not claiming that this is true, but only that the chain of reasoning holds, and therefore this qualifies as a self-consistent, falsifiable statement.
> it would stand to reason that high-IQ people would move up the income bracket scale at a higher rate than they would move down and vice-versa.
How does that stand to reason? We're both speculating here, but it seems to me for that to be true, we'd have to be living in a meritocracy of some sort. And we're not.
Subjectively, I've known lots of people all up and down the income spectrum, and I've never noticed that wealthier people tend to be smarter or that poorer people tend to be dumber. Broadly, it seems to be the same mix no matter what.
What does seem to be broadly true is that you can predict a person's wealth level based on their parent's wealth level.
I do not wish to respond to your comment as our world-views are not compatible and I don't feel like getting into the entire meritocracy discussion at this time.
I'll stay on topic and reply to this:
> How does that stand to reason?
It is sufficient to believe that increasing a person's IQ increases the expected rate of change of that person's wealth. I make no other hypotheses.
IF you accept that, THEN it is a mathematical necessity that there exists a correlation between IQ and wealth.
I believe empirical evidence does support this. We could discuss at length how strong the effect is, but I don't think claiming no such effect exists is tenable, when you take into account e.g. the well-known correlation with education.
> I don't think claiming no such effect exists is tenable
I was too loose in the language of my last comment. I don't think that there is zero such effect. I think the effect is small enough to not be a significant factor. I've tried to include enough qualifiers (such as "broadly") to indicate that.
> the well-known correlation with education.
If you're saying that there's a significant correlation between education and income, I'll agree with that. But we've been talking about intelligence, not education.
This is true, but I see no reason to think that the distribution of intelligence changes much through the different income brackets.