If disagree with it. High income areas are high rent areas. Is a coal mine in West Virginia really "less productive" than a tech firm in SanFran?

Yes, in the economic sense of productivity, it is. Your high income = high rent is the same statement as what I’m saying.

The mechanisms are the same, though. Amazon putting HQ2 in a city causes rent to go up (higher productivity).

A coal mine developing a new, more efficient train to move coal around will also cause rent to go up (higher productivity).

1. You didn't provide any actual evidence or deeper claims than (putting something in parenthesis). I don't really believe the coal mine claim, as in reality we don't see that happening. Mining towns are still among the cheapest and poorest towns in America.

2. That's a really naive sense of economic productivity. Here is a thought experiment.

A) Let's shut down all of Amazon HQ1.

B) Let's shut down the dams that provide 86% of Seattle greater area's electricity.

Which one of these entities is ACTUALLY responsible for that "productivity" then? And I bet you rents in Seattle without electricity would drop far greater proportionally than they do out by the dams. Rent follows incomes, not productivity.

There is no deeper claim being made. We are saying the same thing. You can call it income if you'd like. A person's wages are some portion of their productivity. The distinction is irrelevant for this point.