Is it better to tax goods with inelastic demand? (Atkinson-Stiglitz redux)

Not necessarily.

Bloggers across the political spectrum, from Noah Millman to Ryan Avent, have recently argued that because the demand for oil is inelastic with respect to price, a gas tax is a relatively efficient way to raise revenue. Though I support a higher gas tax, I don’t think this argument is quite right.

It’s true that the deadweight loss from a tax (a standard measure of inefficiency) is intimately connected to the supply and demand of the good being taxed. If either supply or demand is inelastic with respect to price, the deadweight loss will be low. But when designing a tax system, the deadweight loss isn’t the only relevant consideration.

To see why, let’s consider a silly example. Imagine a tax with the lowest deadweight loss imaginable: a tax on breathing, levied at a rate of $10 every day that you take a breath. The demand elasticity here is exactly zero—no one (I hope) is going to stop breathing for an entire day just to avoid paying the tax. This is a wonderfully efficient tax: it has zero deadweight loss.

In fact, this is effectively a lump-sum tax, a tax demanded equally of every individual. Since lump-sum taxes are a perfectly efficient way to collect money, anyone thinking about how to minimize deadweight loss will always raise revenue exclusively through lump-sum taxes. But this is a cheat: obviously we don’t observe lump-sum taxes in practice, even though at the margin it would be possible to implement them. (If the government demanded that everyone coughed up an additional $50, almost everyone could manage to do it.)

The reason why the government doesn’t do this, of course, is distributional: we care about the poor, and it wouldn’t be fair to ask them to pay exactly the same dollar amount as the rich. This is important for even the most conservative fiscal policy proposals—many of them suggest flat taxes, but never a regressive lump sum. And one we realize that this is a key issue, it’s critical for our understanding of optimal taxation, far beyond the observation that lump sum taxes are a bad idea.

A famous result in public finance, the Atkinson-Stiglitz theorem on the optimality of direct taxation, captures this intuition in a striking way. Atkinson and Stiglitz show that if the utility function is weakly separable between consumption of various goods and labor (intuitively, the amount you work has nothing to do with your consumption choices, except insofar as working more gives you more money to spend), then no consumption taxes are needed to attain the optimum. You don’t levy higher taxes on gas than soy milk because soy milk has a higher elasticity; you just determine the optimal tax schedule on income and then stay away.

Why is this? Well, the reason we don’t levy lump-sum taxes is that we’re concerned about distribution. But it’s also a story about information: if we knew exactly what each person was capable of making, we could assign everyone personalized lump-sum taxes, and achieve our distributional goals without distorting any incentives. (“Matt, the government has calculated that you are easily capable of making $1 million a year over the next 10 years; therefore you must pay $300,000 in tax every year, regardless of your actual income.”) Of course, we don’t have this information, and we’re forced to use income as a proxy for ability to pay instead. But then we reach the Atkinson-Stiglitz result: if your exact consumption decisions don’t add any information about this ability, they shouldn’t be part of the tax. Unless gasoline consumption indicates that you have high hidden earnings potential, it shouldn’t be subject to any special charges.

Mapping this result onto the real world is tough, since “income taxes” in practice include taxes on saving (i.e. capital income), rather than the original income itself. Unless propensity to save or access to savings vehicles offer additional information about underlying earning ability, the Atkinson-Stiglitz results suggest that taxes on capital income should be zero, and that our current regime is inefficient. In this context, it might be efficient to raise gas taxes if they’re replacing taxes on capital gains. But it would be even more efficient to replace capital gains taxes with taxes on regular income*, or implement a progressive consumption tax. It’s tough to imagine a world where hiking gas taxes is the fiscally optimal thing to do, or even the second best.

Of course, this isn’t to say that gas taxes aren’t a good idea. If they’re correcting an externality, gas taxes are absolutely appropriate. They’re just hard to justify on fiscal grounds alone.

*Since people with lots of capital tend to be rich, and they acquired that capital under the assumption that it would be taxed, in practice a sudden implementation of this change would represent a windfall gain for the rich. But this could be counteracted either through an explicit one-time expropriation of capital or an implicit expropriation through an increase in consumption taxes.



Filed under fiscal

15 responses to “Is it better to tax goods with inelastic demand? (Atkinson-Stiglitz redux)

  1. wintercow20

    Hi Matt,

    I am on the bandwagon here, mostly. Three points of interest perhaps.

    (1) Given your description of the informational problems inherent in optimal lump sum taxation schemes, you may be interested to read up on an old tax called the “Faculty Tax.” I came across an old AER article, perhaps 100 years old, that described how several colonies used to assess taxes based on potential.

    (2) Given your last comment on Pigovian gas taxes, how would you respond if it were demonstrated that demand for gasoline was (implausibly) perfectly inelastic. Sure, my driving imposes costs on others, but there isn’t a problem of “too much” driving in that world, we would again only seem to be facing a distributional challenge. A Pigovian tax does nothing for efficiency there (and hey, the planet can even be destroyed and it would be “efficient?”).

    (3) Interesting point about windfall gains on capital accruing people. Ignoring the property rights question, such a view would also imply that if we implemented a cap and trade program instead of a tax, it is not clear that permits should NOT be given directly to the heaviest polluters — since they have long operated under the assumption that they would not be taxed for said emissions.

    • (1) Interesting—I’ll check it out. I always assumed that this was only a hypothetical device for economists, not an actual tax proposal at some point in history. (Though I can imagine some cases where a limited form of faculty taxation would be politically plausible: for instance, in countries where a small elite ethnic group has vastly better prospects than the rest of the population, independent of current income. Of course, the long-term political economy consequences of permitting such taxes would be disastrous.)

      (2) In the case where driving was perfectly inelastic, I think that the Atkinson-Stiglitz result would say that taxing it wouldn’t matter: taken alone, it might have adverse distributional consequences, but as long as income is a sufficient statistic for underlying ability (e.g. consumption doesn’t tell us anything more), designing the correct nonlinear tax schedule is sufficient for distributional optimality. And if driving is not perfectly inelastic and imposes an externality, Atkinson-Stiglitz would suggest that we tax it at exactly the level of that externality (though to be certain, of course, I would need to write this up more formally).

      (3) I agree that there is a parallel here. In general, however, I support mild expropriation of polluters’ wealth because I think it provides good ex-ante incentives: the political system always takes much longer than is optimal to get around to taxing some newly dangerous form of pollution, and the inefficiency from this delay may be lessened if polluters anticipate adverse consequences from sinking too much capital into, say, a polluting factory. Moreover, giving cap-and-trade permits to polluters overcompensates them (they will actually make money from the scheme once they adjust their production to the new implicit price and sell the rest) and perversely incentivizes them ex-ante to pollute more to get a higher quota.

    • 2) I don’t know if this is avoiding the issue… but surely if demand for gasoline is perfectly inelastic (yet imposes an externality), a price instrument such as a Pigovian tax is the wrong policy tool to consider from the outset… Rather you should be using a quantity instrument, such as cap-and-trade, which guarantees lower output (and has no distributional inefficiencies).

      3) I’m not sure about this. I thought that point about windfall gain was relevant to the extent that they were associated with the efficiency of fiscal taxation – NOT correcting for an externality. You would still be “taxing” the same underlying negative asset (i.e. pollution) and not two distinct assets (income vs capital).

  2. Scott Sumner

    Excellent post.

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  4. Philipp Heller

    There are further results in public finance which would suggest that taxation of gas is optimal. In a recent article ( in the Journal of Public Economics, L. Kaplow shows that differential commodity taxation is not optimal under the assumption of weak separability (as in the A-S result) and the existence of an income tax, which does not need to be at the optimum. The intuition behind the result is that if one starts by eliminating differential commodity taxes and then adjusts the income tax schedule to keep everyone’s utility at the pre-reform level and at the pre-reform labour supply (which is possibly due to the separability assumption), then the net effect of the reform will be an increase in government revenue, which in a last step could then be given to the consumers, thereby raising their utility.

    Relaxing the weak separability assumption it can be shown that goods which are complements to leisure should be taxed more highly than goods which are substitutes for leisure. As driving around in a car usually requires time, gas will be a complement to leisure (note though, that this would not apply to work-related driving, which should thus not face a higher tax) hence a higher tax rate on gas would be justified.

  5. JPCousteau

    Just a brief point that in some states with local income taxes (the largest example being Pennsylvania in its entirety along with some other cities) They do use a system of lump sum plus percentage taxes on income. I’ve never seen a point where the reasoning for this has been explained, but such taxes do indeed exist, much to the chagrin of kids with part time jobs where the $50 or so makes up their largest non Medicare/Social Security tax.

  6. Explodicle

    Then of course there’s the land value tax, which has inelastic SUPPLY. It’s progressive, simple, efficient, and tested worldwide.

  7. GU

    Phillip Heller is correct that the AS 1976 results don’t hold if the strong assumptions are relaxed. The uniform commodity tax (i.e an income tax) result assumes homogeneous preferences. Once heterogeneous preferences are allowed, differential commodity taxation (including taxing savings) might be on the table. Relative substitutes for leisure should be subsidized and relative complements to leisure taxed, in order to make up for the labor distortion inherent in taxing earnings.

    But identifying relative complements/substitutes to leisure with the needed accuracy for real life implementation is only slightly less realistic than assuming homogeneous preferences. Arguments about the ideal tax system are fascinating, but in the end administrative considerations might dominate. Consumption taxation, particularly of the indirect type like personal cash flow consumption taxes or David Bradford’s X Tax, are superior administratively to our current income tax. These taxes can be just as progressive as the income tax, so the fairness arguments are moot. In fact, the fairness argument is in favor of consumption taxation, since income taxation favors spendthrifts over savers, while consumption taxation (assuming constant tax rates) treats spending and saving the same on a present discounted value basis.

    Saez’s 2001 paper makes an indicator goods argument in favor of taxing savings. Indicator goods are those goods that are overwhelmingly consumed by high ability (aka “faculty” or “endowment”) types (think tickets to the opera). Taxing indicator goods is a way to indirectly tax ability, the goal of traditional public finance (that is assumed to be impossible by most models, included AS 1976). Saez argues that saving is done disproportionately by the highly educated, and the highly educated are disproportionately high ability, ergo we should tax savings as an indicator good. The flaw in his argument should be obvious—it is education we should tax if Saez’s premises are true, not savings. Indicator goods logic also depends on the elasticity of demand for the indicator goods (if the good is not inelastic demanded, it is unlikely to be an efficient good to tax).

    Those interested in this stuff might wish to read the following paper: Joseph Bankman & David Weisbach, The Superiority of an Ideal Consumption Tax over an Ideal Income Tax, 58 Stanford Law Review 1413 (2006).

  8. The gas tax is more administratively efficient than almost anything since it is actually only collected from a few refineries, not 200 million drivers. This should count for something when choosing a tax regime.

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