PRESIDENT BUSH'S proposed tax cut has rekindled an age-old debate: how progressive or regressive should the income tax be?

The United States has a generally progressive income tax, which means that at a given level of income, the marginal tax rate (the rate paid on the last dollar earned) is higher than the average tax rate at that level of income.

The result, according to a Heritage Foundation report by Daniel Mitchell, is that the top 1 percent of income earners account for about 35 percent of personal federal income tax collected, and the top 25 percent of the income distribution pays almost 83 percent. Over all, the upper half of the income distribution pays 96 percent of all federal income taxes collected.

Those who argue for a more progressive income tax emphasize equity: a tax dollar paid by a rich person causes less pain than a tax dollar paid by a poor person. Those who argue for a less progressive system emphasize efficiency: the most productive people should face lower tax rates to give them strong incentives to work harder and produce more.

These trade-offs have been examined in the economic literature concerning the ''optimal income tax.'' Economists model the trade-off as a mathematical optimization problem in which the quantity being maximized is some measure of overall welfare, which typically involves a concern for equity, while the constraints have to do with efficiency issues, like creating appropriate incentives for producers.

This formulation of the optimal income tax problem was first examined by the economist James Mirrlees of Cambridge University, who received a Nobel in economic science for his analysis. In the simplest version of the Mirrlees model, taxpayers differ only in their ability: how much they can produce with a given amount of effort. One striking result of this model is that those at the very top of the income scale should face low marginal rates.

This result emerges from a detailed mathematical analysis, but the intuition is not hard to explain. Let us assume, for the sake of argument, that Bill Gates made $1 billion in 2000, an amount larger than any other American taxpayer. Suppose further that despite the best efforts of his accountants, he ended up paying 40 cents of the last dollar he earned to the Internal Revenue Service.

Consider the following thought experiment: drop the marginal tax rate from 40 percent to zero for all incomes above a billion dollars. The I.R.S. won't lose any revenue from this reduction, since no one has an income larger than $1 billion. And who knows -- the lower marginal rate might encourage Mr. Gates to work a little harder in 2001, producing new products that would make him, and the rest of us, better off.

Of course, the fact that it pays to reduce the marginal tax rate for billionaires doesn't say much about what tax rates should be like for mere millionaires, a point that has been emphasized by Professor Mirrlees himself and confirmed by subsequent researchers, like Peter Diamond of the Massachusetts Institute of Technology and Emmanuel Saez of Harvard. But the intuitive argument presented above is pretty compelling: if income depends only on ability, those at the very top of the income-ability distribution should face low marginal tax rates.

But perhaps this model is too simple. One might well argue that Mr. Gates, as productive as he is, doesn't owe his success entirely to ability: there was a lot of luck involved, too. And, if truth be told, that's probably true even for mere millionaires.

So let's consider a different model: one in which differences in income are a result only of luck and have nothing to do with ability. In this case, the optimal income tax may well involve taxing billionaires at very high marginal rates. True, aspiring billionaires won't work quite as hard, since the after-tax reward from hitting $1 billion has been reduced. But the chances of becoming a billionaire are pretty low anyway, so taxing billionaires at a high rate won't really discourage much effort by those hoping to become one.

Thus a model where luck is the driving force tends to yield a more progressive optimal tax than a model where ability is the driving force. This is about as far as theory can take us, but it highlights the critical question: How much income results from ability and how much from luck?

It is safe to say that this question has not yet been completely resolved by the economics profession. Still, everyone seems to have an opinion about it: if you want to determine whether someone is a Republican or a Democrat, just ask that person whether differences in income come mostly from luck or from ability.

The preliminary evidence available from in-depth surveys like the Panel Study for Income Dynamics at the University of Michigan shows that income varies a lot from year to year for many households. Economists have found that random events like episodes of bad health, accidents, marital dissolutions and family emergencies play a large role in short-run year-to-year fluctuations in income.

A Harvard social policy professor, Christopher Jencks, and his collaborators pointed out many years ago that income inequality among brothers, who share similar genetic and environmental characteristics, is almost as great as for the population as a whole. This suggests that luck is an important factor in the long run as well.

If luck plays a substantial role in the determination of income, it makes sense to have a progressive income tax, creating a form of social insurance in which the lucky subsidize the unlucky. Perhaps the folk singer Phil Ochs had the best answer for why the upper half of the income distribution should pay so much more in taxes than the lower half: ''And there but for fortune, may go you or I.''