In the vain hope of making sense of the president’s budget proposal, I have begun reading it with some care. I’ve become stumped, however, by some of the math used to justify his tax plans. His budget promises, as did his campaign, to only raise taxes on the top 5% of American households or those couples making over $250k ($200k for individuals). My first problem is that those two cutoffs are not the same.

According to the Census Bureau’s current population survey, the top 5% of households in 2007 were those with incomes over $177,000. Thomas Piketty and Emmanuel Saez’s data on tax returns (the Gold Standard for measuring the income distribution) show that the cutoff for the top 5% was $148,000 in 2006 (I believe their number is lower because they exclude government transfer payments.) So my puzzlement is whether ultimately Obama’s commitment is to not raise taxes on those families below $250k or is to not raise taxes on those below $177k or ($148k).

The question has both important political as well as economic ramifications. The Republicans will obviously fare much better in their quest to defeat Obama’s tax plan if they can portray Obama as ultimately willing to extend his tax increases to much lower income levels.

The economic ramification is one that exacerbates the political one. To pay for the spending increases that the budget calls for and to keep deficits within reason, taxes will almost certainly have to go up on families making more than $150k (if not less).

A little of my math helps explain why I believe this. Assume that tax increases are limited to those families making above $250k. From eyeballing Picketty and Saez’s data, 33% is a reasonable estimate of the share of national income received by this group of families. This implies that to increase revenue by 1% of national income the average tax rate on those above $250k will have to rise by 3 percentage points (because they control a third of national income their taxes must go up three time the total increase). Obama’s budget projects deficits at 4% of GDP from 2013-2019 (even after an initial round of high-income tax increases). So if these future tax liabilities are to be borne only by the $250k-and-up club, their taxes would have to go up 12 percentage points. How large a tax increase would this be? Currently, the average federal tax rate on families over $250k is about 30% (this includes all federal taxes). Obama’s proposed increases are likely to push that number to around 34%. To cover all future liabilities only through taxes on this group, their average tax rates will have to rise to 46%, a 50% increase in federal taxes. By contrast, the rate on the top 1% was 38% in the last year of the Carter administration before Reagan’s tax cuts. Even if Obama were willing to raise taxes on the top 5 percent (i.e. those with incomes greater than $150K), the average tax increase for that group would still be around 10 percentage points. While I often believe that supply-siders overstate their case against taxes, this clearly represents a level at which perverse incentive effects including non-compliance and reduced labor supply (especially among second earners) could materialize. So even tax hikes of the magnitude I’ve described may not produce the desired level of revenue.

So it seems very clear that enactment of Obama’s ambitious spending plans (many of which I support) will require taxes to go up (now or later) much more broadly than either his $250k or top 5% pledge would seem to allow. An administration that prides itself on its openness and transparency would do well to acknowledge that inconvenient fact sooner rather than later.