This delightfully playful and whimsical paper formed the basis for a fun and interesting discussion at the colloquium today. A notch (the paper's topic) in the narrow sense arises when a tax or other fiscal system features gives rise to a discontinuous jump in taxes or transfers at a given point as the quantity of something (such as income) rises. Thus, suppose we enacted a 10% income tax that only applied to individuals earning $100,000 or more. If it applied to ALL of such individuals' income, rather than having a $100,000 exemption amount, then earning one more cent so your income went from $99,999.99 to $100,000 would cost you $10,000 of tax. Pretty steep marginal tax rate on that penny.
While that example may sound silly, there actually are features like that in various fiscal systems. E.g., as Slemrod explores more fully in a related paper, "Car Notches," the gas guzzler tax on car manufacturers has that structure a bit. So do value-added taxes (VATs) in some countries with small-business exemptions that disappear without a surviving exemption amount once the business crosses a given size threshold. And Medicaid has that character in the U.S. - earn too much, and you lose this entire, rather valuable benefit all at once (or actually not quite - I believe the notch effect is mitigated slightly by time lag rules for losing Medicaid eligibility).
I'd define a notch in this sense, in the clearest base case, as existing whenever the marginal tax rate at a given point is greater than 100% or less than 0% (on the latter, suppose Medicaid eligibility were only for non-poor individuals who earned enough - not that this would make much sense, but just to illustrate the point).
One distinctive move the paper makes, which gave us plenty of grist for discussion and criticism but was conceptually a move worth making, was to compare these relatively clearcut "quantity notches" to "characteristic notches" that arise from line-drawing. E.g., an instrument is somewhere on the continuum between being classified as debt or as equity. One iota more towards the debt side of the spectrum, and its character for tax purposes discontinuously changes - it now qualifies (100% rather than 0%) as debt.
Or geographically, I'm in NYC and face the NYC sales tax. But I'm walking towards the Connecticut border. I step across the NYC line into Yonkers, and with that one step (supposing for convenience that the tax legally depended on where I made the purchase, rather than on where I consume the good), and my sales tax rate drops by several points. When I finally make it to Connecticut, with that last step my tax rate drops still more. So there are geographical notches in sales tax liability in this example, although it's hard to express them as marginal tax rates (since what would be the denominator).
In the discontinuity discussion, I added some non-tax examples, such as the negligence standard for tort liability, in which taking just barely too little care leaves you liable in full for the harm caused, whereas an iota more care may leave you completely non-liable. As discussed in a well-known article by Craswell and Calfee, at least with uncertainty about appropriate care levels (in the mind of a subsequent fact-finder) this creates peculiar incentives to take too much care when you are close to the line of escaping liability.
The paper takes advantage of the conceptually clearest example, a Pigovian tax in which the correct marginal rate depends on marginal harm caused by increasing scale of the activity. It's difficult to believe a perfect Pigovian tax (say, on pollution) would have notches, since that would require a very bizarre marginal harm structure. But in the Pigovian world one can perhaps rationalize notches as a response either to (1) the costliness or impossibility of continuous measurement or (2) the unavailability of continuous outcomes - e.g. only one of the rival candidates can win an election, to borrow another of my discontinuity examples.
The harder question was how well the Pigovian analysis applies to the non-Pigovian setting. E.g., would it be desirable (for New York? Or for a benevolent higher-level decision-maker interested in social welfare?) to wear away the notch a bit by having the tax rate decline as one approaches the Connecticut border. This reduces the incentive to cross over into Connecticut just for tax reasons, but newly introduces distortions within the now tax-varying jurisdiction.
Also unclear how well the analysis from Pigovian taxes crosses over to, say, debt vs. equity line-drawing, or economic substance rules for tax-motivated transactions (where having "just enough" substance discontinuously switches the taxpayer from losing to winning), or the use of an income tax that inefficiently (even if on the whole optimally, given distributional goals and limits on available instruments) discourages work and saving. But it wasn't the paper's job to try to solve these issues.
The one issue I had on my list that we didn't get to is how continuity should affect our thinking about distributional issues. I had noted that treating the Pareto standard as distinctively important creates a discontinuity in one's social welfare function - eliminating the last scintilla of harm to the last uncompensated loser from a policy change shifts the normative verdict from thumbs down to potentially thumbs up - no matter how great the positive consequences on the other side of the ledger - if one accords Pareto changes special status. Louis Kaplow argues that accepting Pareto plus the principle of continuity means you have to be a welfarist (and perhaps, after arguing with him for a further 20 minutes, a utilitarian).
I'm sympathetic to this line of argument, but others say: Even if I accept Pareto, you can't force me to accept continuity. If I see nothing wrong with letting trivial differences in the outcome radically change the normative outcome, there's nothing you can do to argue me out of it - unless you force me to grant further assumptions that I decline to make. Essentially, the problem is that continuity, asserted for its own sake rather than as an aspect of a given framework (e.g., utilitarianism, or the Pigovian tax approach to efficiency), is a postulate not a theorem. So, however unreasonable rejecting it might appear to some of us to be, you can't force it on the rest any more than you could logically compel them to accept transitivity.
I'm reminded of the hilarious passage in Lewis Carroll's discussion of Achilles and the Tortoise. Achilles gets the Tortoise to accept (i) A, and (ii) if A, then B. But the Tortoise nonetheless declines to accept B. Achilles says: You HAVE to accept it as a consequence of the other assumptions! The Tortoise replies: If that's so, why don't you tell me to accept (iii) If I accept A and also the statement if A then B, then I must also accept B.
Now Achilles thinks he's got him - the Tortoise HAS to accept B now. But the Tortoise replies: I don't see it. But if you like, I'd be happy to accept proposition (iv), which holds that, if I accept the first three propositions, then I must also accept B.
Achilles is slow to realize that this won't help either - because if the Tortoise now is compelled to accept B, that point is worth stating as proposition (v). And on that approximate note, Lewis Carroll leaves them to continue their squabble.
Not quite on point, of course, as the Tortoise rejects basic principles about how to reason logically (which he effectively posits are themselves postulates), rather than rejecting propositions (i) or (ii) themselves. But to one who accepts the postulates, rejecting them is comparably baffling.
I'm with Achilles and Kaplow. But then again, I suppose I started out that way - they didn't actually persuade me, so much as express a viewpoint that I share. Accepting Pareto plus continuity leads to welfarism, but you can't logically force someone to accept either of those principles, if they don't independently resonate.