U(X(L)-L-t)/U(X(L)-L) = K
The idea is that the fraction of utility one keeps after taxes is the same for everyone. X is consumption; the amount one gets to consume is a function of effort, L. To make things easy, we will assume people consumer their incomes, so that income and consumption are the same. Assume that utility function has the shape U' > 0 and U" < 0. K is dependent on how much society wishes to spend on public goods.
Just this simple formulation presents three problems. First, the fair rate of progressivity will be a function of the magnitude of U". For instance, if we assume log utility, U' = 1/(X-L) = U" = -1/(X-L)^2. This means U" gets very small very rapidly, which also means that the need to increase marginal tax rates in income to maintain the above definition of fairness gets quite small. We do know that taking money away from people at or below subsistence levels of income will lead to substantial diminution of utility, but beyond that point it is hard to say how sharply progressive taxes need to be in order to be fair.
Second, the correspondence between consumption and effort is not one-to-one. If the correlation between consumption and effort is less than one--and I will go out on a limb and say that it certainly is--taxing income actually only approximates taxing utility. The lower the correlation, the worse the approximation.
Finally, defining effort is a problem. As Matthew Yglesias notes, NYU professors make a lot less money than Wall Street bankers, but their life might well be better. Perhaps I am wrong, but it seems to me that the -L in a steel worker, coal miner, or line worker is a lot bigger than mine, and so looking at income alone is adequate for approximating utility.
So what to do? Here is why, despite my liberal leanings, I find a flat tax with a large exemption and a large earned income tax credit appealing. The rate would have to be sufficient to raise revenue, and would apply equally to all type of income. Deductions would be limited. Such a set up would assure that Warren Buffett would pay no less a share of his income than anyone else. Bob Hall proposed a similar plan 15 years ago. I would dress it up with the earned income tax credit.