Last week, I challenged readers to reconcile two apparently contradictory statements, both of which are frequently made in economics textbooks:
- To minimize distortions, all goods should be taxed equally.
- To minimize distortions, inelastically demanded goods should be taxed more heavily. (This is sometimes called the Ramsey rule, after Frank Ramsey, who plays a major role in the final chapter of The Big Questions).
I’ll give you the answer in a minute. The executive summary is that a) “Inelastically demanded goods should be taxed more heavily” is true only in very special circumstances; in general a much more complicated formula is needed, b) When all goods can be taxed, that complicated formula does in fact tell you to tax them all equally, and c) a lot of textbooks give incredibly misleading accounts of all this.
The more detailed answer follows; if you prefer a more mathematical account, click here. To keep things manageable, I’ve assumed all supply curves are perfectly elastic.
First, “inelastically demanded goods should be taxed more heavily” is a gross oversimplification of the Ramsey rule. To see why it can’t be correct as stated, suppose you demand only two goods: Peanuts and root beer. You use your $20 income to buy 3 root beers, regardless of the price (at least as long as you can afford them) and spend the rest on peanuts. Then root beer is inelastically demanded. But it’s crystal clear that as far as your behavior goes, a tax on root beer and a tax on peanuts are equivalent — either way, you’ll keep right on buying 3 root beers plus as many peanuts as you can afford. In other words, no rule that says it’s better to put a higher tax on root beer can be correct in this example.
The correct statement of the Ramsey rule places heavier taxes not on the goods that are inelastically demanded, but on those goods whose prices lead to the fewest distortions (in a sense that can be made precise) not just in their own markets but in others as well. If the price of root beer affects your peanut consumption, that goes into the calculation. If the price of butter affects your bread consumption, so does that.
If you’ll forgive a little jargon, then, the correct statement is that the optimal mix of taxes depends on a very complicated formula involving not just own-price elasticities but cross-price elasticities. Only in very special circumstances does this reduce to the cartoon version that says “inelastically demanded goods should be taxed more heavily”.
Second, once you write down the correct (quite complicated) version of the Ramsey rule, you discover that as long as all goods can be taxed, the Ramsey rule does tell you to tax them all equally. Thus it gives the same answer as the usual argument using indifference curves.
Third, if only some goods can be taxed, then it’s not in general optimal to tax them all equally. The Ramsey rule (correctly stated) tells you how to tax them.
Fourth, leisure counts as a good. If you can’t tax leisure (or equivalently subsidize labor) then it’s not in general optimal to tax everything else equally. However, if labor is supplied inelastically (as the labor economists tell us is more or less the case) then the tax-everything-equally result is restored, even when leisure can’t be taxed.
And fifth: A great number of elementary textbooks either get this wrong or present it so misleadingly that it might as well be wrong. Students beware.