The other day I tweeted:

Under a Tiebout federal government, I posit the rich would pay more in dollars than the middle class, less as a percentage
Tiebout is shorthand for “voting with your feet,” and it’s a way to think about local governments competing against each other.  If citizens and firms could relocate easily, governments that offered low value for the tax dollar would lose citizens, and those that offered good value would gain citizens.  
Highly progressive taxation seems like a bad deal for high earners, so I wouldn’t expect it to survive competition.  If McDonald’s started a progressive pricing policy, I’m guessing high earners would switch to another restaurant pretty quickly.  The same probably would be true for competitive governments.  
So, is my hunch right? U.S. state governments might be a decent proxy for Tiebout governments–it’s fairly easy for most people to move across borders if they don’t like the price/service bundle they’re receiving.  Fortunately, public finance researchers have estimated these numbers repeatedly.  Here’s a graph from a report by the Institute on Taxation and Economic Policy.  The light blue sums up all state and local taxes, not just income taxes: 
An academic paper (gated) by Chernick with similar findings for other years is here
I should note though, that while state taxation is regressive in percentages, it’s progressive in dollars.  And that’s the point of my tweet: Higher earners pay more than lower earners even though they could leave.  Perhaps some of that is altruism, but I suspect-without-proof that most of it is just that the rich (and middle class) buy more and better government services than the poor.  
It’s possible that progressive income taxation could coexist with voluntary competitive government. Maybe the high-skilled need to be near each other to produce a lot, so the locales preferred by the rich can tax that demand for proximity.  The (not very progressive) New York City income tax comes to mind.  

But at the national level, I suspect that the reason the rich pay higher total tax rates is mostly because it’s hard to leave the nation.  Easy targets.  

At least for the U.S., when people can shop for their own social contract the equilibrium tax system is usually progressive in dollars and regressive in percentages. It’s worth keeping this concrete collection of social contracts in mind when hypothetical social contracts are being debated.