Do the Poor Prefer the Rich to be Idle?

Matthew Yglesias is disheartened by my need to praise his theory:

Meanwhile, I was sad to read this comment from Andrew Samwick on my theory that tax policy affects pre-tax income distribution:

I hadn't fully appreciated that a progressive tax system might be used to give lower-income workers a leg up in competing for the marginal unit of production. It remains an empirical question as to how important this might be over the last 25 years. I would have thought the effect to be small, compared to things like increasing global competition in product markets.

I would have thought that lots of people would have thought of that already. It doesn't seem like it should be too hard to model, right? Indeed, at first I thought I should be able to model it myself but actually I can't. Some of what I'm thinking, though, goes below.

Here's some clarification of what I meant by "fully appreciated."

Progressive tax systems reduce the inequality in the after-tax distribution of income, conditional on the pre-tax distribution of income. Got it.

High marginal tax rates on the rich discourage further work by the rich, lowering their pre-tax income. Got it.

With the rich discouraged, there is more demand for the poor to do that work. They work more, and their pre-tax income goes up. Hadn't thought about the impact of the lower marginal tax rate for poor people on their willingness to work to meet that extra demand until I read Matt's theory.

So here's the next question, which has been asked in different ways by different people: are the poor better off if the rich are idle? I suppose they could perceive some direct benefit of a lower income gap, but I wouldn't presume. The answer, I think, depends on how the rich and poor interact in production and consumption. What if Tiger Woods plays one fewer golf tournament? What if Paul Krugman writes one fewer book? What if Bernie Marcus had opened a hundred or so fewer Home Depots?

It's the last one--the effect on entrepreneurs--that might make us think twice about the answer.