Monday Puzzle: The Least Bad Tax

Today’s puzzle is specifically for the econo-geeks. Less geeky fare will follow in the near future.

Two of the main lessons that our undergraduates typically take away from their introductory classes are these:

  • To minimize distortions, all goods should be taxed at the same rate.
  • To minimize distortions, inelastically demanded goods should be taxed most heavily.

What is the correct response to this pair of apparently contradictory lessons?

  1. Economics is large. It contains multitudes. Get over it.
  2. Nobody ever said that all goods should be taxed at the same rate. The right lesson is that goods should be taxed in such a way that their prices all rise by the same ratio. That’s not the same thing, because different taxes are passed on to consumers in different proportions. So maybe that’s somehow consistent with the elasticity rule.
  3. Both of these “lessons” are based on maximizing consumer welfare. Maybe if we account for the producers, somehow everything will magically make sense again.

The right answer, alas, is “none of the above”. A is a non-starter. As for B and C, let’s assume that all supply curves are horizontal, in which case all taxes are passed on to consumers (which dispenses with B) and producers earn no surplus in any event (which dispenses with C).

So how about this?

  1. My Principles teacher never said anything about setting tax rates so that all prices rise by the same ratio! You’re just making that up!

Alas, what’s going on here is that your Principles teacher either didn’t cover indifference curves or didn’t have time to cover all of their cool applications. It is in fact true that to minimize distortions (at least among consumption choices, which, in view of our assumption about horizontal supply curves is all that matters here), all goods should be taxed in such a way that their after-tax prices rise in the same proportion. In other words, if we’re deciding between a “blue tax plan” that raises the price of eggs by 6% and the price of wine by 4%, and a “red tax plan” that raises both prices by the same amount, and if the rates are calibrated so that both plans raise the same amount of revenue for the government, then the representative consumer prefers the red tax plan. The usual argument is via indifference curves and hence doesn’t always make it into the Principles courses, but if you study economics long enough, you’ll run into it eventually.

Let’s try again:

  1. Hey, wait a minute! Your indifference curve diagram assumes I can only tax consumption goods. Shouldn’t I also have the option of taxing labor and/or leisure?

You’re right. The indifference curve diagram takes income as exogenous. If you’ve got to earn your income, then we should include leisure as another good. Go ahead and do that if you like, or just continue to assume exogenous incomes. Like options A, B, C and D, this is still a blind alley.

So what’s the right answer? It took me a little time to settle this in my own mind. (The question first popped into my head back when there was still a race for the Republican presidential nomination and I was trying to figure out if there was any way you could possibly justify Rick Santorum’s manifestly insane call for preferential taxes on manufactured goods.) Once I’d understood it, I Googled around a little and was surprised to find nothing on the Web that clearly addressed and resolved this little puzzle. So I thought I’d invite my readers to tackle it and see if their answers coincide with mine.

Please don’t post spoilers in the comments if you’re an expert in tax policy and have understood all of this forever, but all others are welcome. More than one very bright econ prof has tripped over this.

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65 Responses to “Monday Puzzle: The Least Bad Tax”


  1. 1 1 Categories+Sheaves

    Red tax plan versus blue tax plan? I see what you did there…

  2. 2 2 Mike H

    ok, let me see….

    if demand is inelastic, that means … um … that if the price of that good goes up, and the price of others doesn’t change, then demand doesn’t change.

    Am I right so far?

    hmm… I’m trying to draw some indifference curves, with budget constraints (see? I’ve learned something from your blog!) if I change P1, I can only get Q1 to not change much if the the budget constraint and indifference curves are almost parallel to the Q1 axis.

    Hm.. This means P1 is very small compared to P2.

    Am I getting closer?

  3. 3 3 Harold

    Last time we had a paradox it was resolved by the difference between willingness to accept and willingness to pay. Don’t see how that helps here.

    To clarify, are we accepting that the first proposition is wrong: to minimise distortions all goods should not be taxed the same, but be taxed so the prices rise by the same proportion. Is this correct? We are looking for a mechanism whereby inelastic goods need to be taxed more in order to raise prices by the same amount?

  4. 4 4 Steve Landsburg

    Harold: Yes, we are accepting that the correct statement is that goods should be taxed in such a way that their prices rise by the same proportion. However, we’re also simplifying the problem by assuming all supply curves are horizontal, so that this correct statement is equivalent to the proposition that all goods should be taxed at the same rate.

  5. 5 5 Conlan

    …definitely geeky, Monday stream-of-consciousness prose.

    Stating your fundamental question in one direct sentence would be helpful.

    ‘Least Bad Tax’ from what perspective ??
    {college students, economics professors, taxpayers, politicians, taxi-drivers, Chinese peasants, etc ??}.

    ____

  6. 6 6 Ken B

    Not being an econ-geek myself I do have a thought/question.

    Say total income is very large and the tax increase very small (we live in Eceerg). Then putting all the tax on an inelastic item will to a first approximation work just fine. Only when the total effect of the income loss becomes significant will discrepancies get big. Kind of like a Geffen goods effect. I think I am assuming a positive savings level here. So does adding savings to the mix help?

  7. 7 7 nobody.really

    I’m really intrigued.

    If the goal is to collect $X in revenues while minimizing distortions in economic behavior, I assume we’d collect all the money from inelastic sources (a Ramsey tax). If I recall correctly, Landsburg suggested a head tax – on the theory that our demand for retaining our heads was fairly inelastic.

    If we’re not opposed to distorting behavior in socially-valued ways, we might also raise revenues by taxing things we want to discourage (a Pigou/excise taxes).

    I sense we don’t rely entirely on such efficient taxing strategies because our concerns extend beyond merely minimizing economic distortions, or even minimizing inefficient economic distortions. We’re also concerned with growth, equity, wealth distribution, the administrative challenges associated with extracting blood from turnips, etc.

    I understand Landsburg to ask for a strategy by which we can ensure that all goods are taxed at the same rate (or that taxes produce equal percentage increases in all taxed goods), and that we tax inelastically-demanded goods more heavily than goods facing elastic demand. Even without knowing anything about taxes, the problem seems to ask for an impossibility. How can we tax all goods equally while ensuring that we’re taxing some goods more heavily than others?

    I guess we could set the tax at 0%. In this fashion we raise the tax on all good equally, while also increasing the tax on inelastically-demanded goods at many multiples of the increase on elastically-demanded goods.

    Alternatively, perhaps there’s a way to alter people’s elasticity of demand such that, at the margin, the elasticity of everyone’s demand for everything is equal?

    I suspect I’m not really understanding the question.

  8. 8 8 Ken B
  9. 9 9 nobody.really

    Ok, here’s an observation:

    As you increase taxes, even if the rate of taxation/percentage price increase remains constant for all goods, government will receive an ever-larger share of its income from taxes on goods for which the demand is inelastic.

    (Raise taxes high enough and government receives revenues solely from the sale of inelastically-demanded goods; all other sales have been squeeze out due to income effects.)

    Does this help?

  10. 10 10 Patrick

    It seems like there might be two different types of distortions..

    •To minimize distortions, all goods should be taxed at the same rate.
    I think this distortion is to utility, as shown by the indifference curves in the OP. Thus, this maxim tries to minimize the distortion to the consumer.

    •To minimize distortions, inelastically demanded goods should be taxed most heavily.
    I think distortion in this sense means to affect quantity demanded. By the nature of inelasticity of demand, those goods do not have good substitutes, because if they did, you could choose the substitute if the price goes up in which case quantity demanded would go down. If you raise the tax on these goods, the price to the purchaser would go up, but the quantity demanded and the price to the producer stay the same. Like good ol’ specialty RX drugs. So I think this minimizes distortion to suppliers.

  11. 11 11 math_geek

    A completely inelastic good combined with another good would make it irrelevant what the taxation distribution is. The utility curves would look like a bunch of points and “space” around the quantity assuming no negative boundary (that is, that the person would always be able to afford his declared preferred quantity). Tax one good or the other, the change always comes out of the non-fixed good. This leads me to believe that it is not so much optimal to tax the inelastic goods as it is that in a world with mostly inelastic goods the taxation does not matter when looked at with indifference curves, which seems like an inherent contradiction between an indifference curve model and a supply/demand model.

  12. 12 12 Tristan

    We want the solution to the bullet problem darn it!

  13. 13 13 khodge

    At first blush it appears that macro and micro economics are being confused.

  14. 14 14 nobody.really

    We want the solution to the bullet problem darn it!

    Another entry in the list of Famous Last Words.

  15. 15 15 Advo

    As nobody.really says, I don’t think I entirely understand what the question is supposed to be.

  16. 16 16 js

    You’d like to be able to raise revenue without changing relative prices – this would be like a lump sum tax, and would be first best. But you can’t achieve the first best outcome in the Ramsey problem, because the planner is assumed to not know an individual’s potential income.

    Suppose leisure was just like any other good, and the planner could observe and tax the total value of leisure a person consumed. Then we’d be in the “don’t change relative prices” world. If the planner can only observe labor supply, then we’re in the “lower taxes on elastic goods” world.

  17. 17 17 David Youngberg

    The question as I understand it is: which tax regime would minimize deadweight loss? (The answer is technically neither; only zero taxes would do that, assuming no externalities, etc. But let’s assume the government wants to collect taxes.)

    On one hand, we want everything to be taxed the same. If the government taxes Andy’s Apples but not Zach’s Apples, then Andy makes no money. Andy’s 1st apple is cheaper to make than Zach’s 10th but no one wants Andy’s apple–it’s more expensive. So Zach sells more, his prices rise, and eventually people start buying Andy’s low hanging fruit. The price of apples is higher, much higher than if a smaller tax would affect them equally. (Imagine Zach not picking the 100th apple and Andy, instead, picking the much easier to get 40th apple.)

    This works find because all apples are identical. Perfect substitutes. There’s a disaster because everyone floods to Zach, supporting his picking of inefficiently located apples. But what if the goods weren’t perfect substitutes? Say cigarettes and apples? Now there are many substitutes for apples, but few substitutes for cigarettes. The demand curve for cigarettes is more inelastic than apples. To keep all things in equal weight, you’ll have to tax cigarettes (and all cigarette substitutes) more.

    In sum, ‘equal tax’ implies the goods are identical. When they are not the same, we adjust for natural differences with different taxes, resulting in the same result if the goods were identical.

  18. 18 18 Scott

    Here’s my interpretation: The two lessons are only contradictory if they’re assumed to apply to the same situations. They don’t. If we have perfect knowledge of the elasticities for all things that might be taxed, we can minimize distortions by making the tax inversely proportional to elasticity, so the second lesson is true in the idealized world of economic models.

    In the real world, we don’t have anything close to perfect knowledge of elasticities for even a small number of things that might be taxed. Given government’s knowledge limitations, the best we can do to minimize distortions is to tax everything at the same rate, perhaps with exceptions for goods where we think we know something about elasticity.

  19. 19 19 Mike H

    Suppose demand for good A is perfectly inelastic, and demand for good B is not, and we tax A but not B. Note that consumption of B will decrease, even though there is no tax on B.

  20. 20 20 Mike H

    Harold: Yes, we are accepting that the correct statement is that goods should be taxed in such a way that their prices rise by the same proportion. However, we’re also simplifying the problem by assuming all supply curves are horizontal, so that this correct statement is equivalent to the proposition that all goods should be taxed at the same rate.

    Umm… so what, exactly, is the puzzle?

  21. 21 21 rednecktech

    In my opinion, most commenters are missing one of the crucial problems with differential taxation. When different items are taxed at different rates to serve some purpose, someone must decide which items get which rate. That someone will, by definition, be a person with authority. That authority is almost never a disinterested party with perfect information. Therefore the differential taxation decisions will be made by someone (or someones) with either a partisan axe to grind, imperfect information, or both.

    The goods that are taxed at identical rates will have some inequity, but the variable rates will will eventually lead to lobbying and other market distortions.

    The greatest good for consumers and producers in the long run will be identical rates.

  22. 22 22 Steve Landsburg

    Mike H:

    Umm… so what, exactly, is the puzzle?

    The puzzle is:

    a) We agree that all goods should be taxed at the same rate.

    b) We agree that inelastically demanded goods should be taxed more heavily.

    c) If one good is taxed more heavily than another, then they are not both taxed at the same rate.

    d) Therefore it appears we are being inconsistent. How do you reconcile this apparent inconsistency?

  23. 23 23 Mike H

    Is it just that if demand for Good 1 is inelastic and demand for Good 2 is not, and the tax department arranges for prices on all goods to increase by 10%, consumers still buy the same amount of Good 1, but less of Good 2, and so the tax dept gets more money from taxes on Good 1, that is, “Good 1 is taxed more heavily than Good 2″?

  24. 24 24 nobody.really

    The puzzle is:

    a) We agree that all goods should be taxed at the same rate.

    b) We agree that inelastically demanded goods should be taxed more heavily.

    c) If one good is taxed more heavily than another, then they are not both taxed at the same rate.

    d) Therefore it appears we are being inconsistent. How do you reconcile this apparent inconsistency?

    Mark Twain remarked, “It ain’t those parts of the Bible that I can’t understand that bother me, it is the parts that I do understand.”

    Analogously, I thought I was struggling with a problem I didn’t understand. But my circumstances are worse than I’d thought: I’m struggling with a problem I DO understand.

  25. 25 25 David Grayson

    Stylesheet fail! I was reading your blog post in Google Reader and the “A.” showed up a “1.”, the “B.” showed up as a “2.”, an the “C.” showed up as a “3.”.

  26. 26 26 Marek

    This will not explain much of economic intuition here, but the difference is this:

    “inelastically demanded goods should be taxed more heavily”: This is a very specific result, in general equilibrium it holds only if the utility is linear in leisure (no income effects) and is additively separable.
    “all goods should be taxed at the same rate”: this, on the other hand, holds for utility that is weakly separable in leisure and homothetic in all the consumption goods.

    I suppose one could prove that if a utility is homothetic, additively separable and linear in leisure then the price elasticities are all the same.

  27. 27 27 Harold

    There seemed to be two broad ways to reconcile. Initially, you could have said that the distortions were not the same, so there is no contradiction. SL has removed that option with the re-stated problem (we agree that everything should be taxed at the same rate and we agree that inelastic goods should be taxed more heavily – no mention of distortions).

    The other is that “same rate” and “more heavily” are not contradictory. Mike H may be on to it. Can we resolve that things taxed more heavily can be taxed at the same rate by differentiating between individual items and classes of items?

    My problem is that we have quite a full explanation of why everything should be taxed at the same rate with the indifference curves. However, I do not really know why inelastic goods should be taxed more heavily.

  28. 28 28 Joseph Zaccardi

    Why not think of the rate of tax as the amount by which consumption in a good is reduced by the population for a given tax.

    For instance, say there are only two goods in the economy, A and B. A is much more inelastic than B. Policy makers should try to set taxes so that consumption in both goods is reduced by an equal proportion.

    As an example, suppose a 15% tax on A and a 5% tax on B reduces the consumption of both A and B by 10%. In this case, the more inelastic good A is taxed more heavily than B, and both A and B are taxed at the same rate as measured by the amount that is consumed.

  29. 29 29 Saturos

    Perhaps if the model in the link showed separate elasticities for the 2 goods (separate income elasticities) it would be possible to resolve the conflict.

  30. 30 30 Arthur Cecil Pigou

    Taxes should be set in such a way that external costs (and benefits!) are passed on to the user, regardless of the distortions this creates.

    So, for example, fossil fuel should be taxes in such a way that the government can afford to clean up any environmental damage caused by fossil fuel.

    Anything else is a subsidy of the distortion.

  31. 31 31 Steve Landsburg

    Arthur Cecil Pigou: Your comments are of course irrelevant to the problem, which presumes no externalities. In the presence of externalities, you’d be relevant but wrong.

    Your namesake promulgated this fallacy about 90 years ago, and it survived for 40 years before it was debunked — but that was over 50 years ago. In other words, it’s been dead longer than it was alive. How odd that it keeps popping up.

  32. 32 32 Bearce

    They’re not reconcilable. Uniform taxes don’t lead to optimal solutions ever (or I should say pretty much ever) if a supply curve is perfectly elastic.

    The optimal taxation on all commodities is one that reduces Hicksian demand curves for goods/services by the same amount.

  33. 33 33 Michael

    Seeing the name of Arthur Cecil Pigou in the thread points to the best tax. :)

    However, why look at at taxing that which is inelastically demanded, and instead look at that which is inelastically *supplied* — i.e. Georgist taxes: http://en.wikipedia.org/wiki/Georgism

    For example, taxing site value does not cause deadweight loss, since the supply of land does not reduce the quantity available. You can theoretically tax it to the point of abandonment without discouraging productivity.

    One could perhaps imagine a system where people competitively bid for sites (the amount to be paid to the government). If a site changes hand, the person winning the bid would also pay the current occupant the fair market value of all improvements on the land, and then take possession.

    Or government could do like they do with property taxes now, and just ignore buildings and improvements when they levy taxes.

    (I know that this is a bit late, but one comprehensive tax proposal I rather like was made by Frank de Jong of the Green Party of Canada: http://frankdejong.blogspot.com/2010/03/green-budget-for-canada.html )

  34. 34 34 Michael

    Oops, I thought faster than I typed. It should read:

    For example, taxing site value does not cause deadweight loss, since taxing land (in the economic sense of natural resources of fixed supply) does not reduce the quantity available.

  35. 35 35 Martin-2

    khodge – “At first blush it appears that macro and micro economics are being confused.”

    Answer A then?

  36. 36 36 vik

    The question as posed is how do we keep price ratios constant while taxing inelastic goods more heavily.
    But surely this isn’t the question we should be asking. Instead we should be aiming for the price ratios that would have obtained if the incidence of the tax was as the Govt wanted (i.e. what would have been the price mix after the Govt’s desired lump-sum tax had been confiscated from everybody.) In other words, some complicated system of subsidies and quotas which has the effect of changing the price ratios for producers to something different from the price ratios for consumers. Needless to say this sort of tinkering is a terrible idea. India actually embraced it in a big way not because its Economists were stupid but because they were just so darn clever- which comes to the same thing.
    The tragedy is that once you’ve got these darn clever guys in charge, especially if they are seen as not corrupt and having bleeding hearts for the poor, then taxes gain legitimacy. But the only sorts of taxes that can be levied sustainably are taxes on goods in inelastic demand . Rents can’t be taxed sustainably because factor mobility is actually pretty high in the medium term. But the income effect of the taxes on goods with inelastic demand can bear down so heavily that substitution effects are swamped and so allocative efficiency appears more and more like some sort of academic pipe=dream and dynamic efficiency takes center stage.
    There is another approach- probably not something sufficiently tractable to be teachable, unless there’s a cool Matematica app for this- which amounts to artificially equalizing elasticities for tradable goods and services- but again, this sort of thinking has already been tried by the good people of India Inc. and we all know how that’s working out.

  37. 37 37 Mike H

    @Marek I was indeed thinking that it would be nice to know the derivations of the two results, and what assumptions lie behind them.

  38. 38 38 Steve Landsburg

    Bearce:

    Uniform taxes don’t lead to optimal solutions ever (or I should say pretty much ever) if a supply curve is perfectly elastic.

    What’s wrong, then, with the standard argument (linked to in the post) that says that uniform taxes *do* lead to optimal solutions when supply curves are perfectly elastic?

  39. 39 39 David

    Maybe the problem is with your words “to minimize distortions” Could it not be that on one hand we want to tax inelastic goods, but on the other we want to tax all goods by the same ratio and that it is trade off we need to balance?

  40. 40 40 Mike H

    Ok, I assumed two goods, consumed in quantities x and y, with utility x*y. I get the same answer with utility log(x*y).

    Consumer prices of x and y are px*(1+tx) and py*(1+ty), wjere tx and ty are the tax rates.

    budget constraint is M = px*(1+tx)*x + py*(1+ty)*y

    The consumer chooses x=M/2/px/(1+tx) and x=M/2/py/(1+ty)to maximise utility.

    tax revenue must satisfy tx*x+ty*y=T. We choose tx and ty to maximise utility, subject to this constraint. This gives

    tx = (py*M-px*M+2*px*py*T)/(py*M+px*M-2*px*py*T)
    ty = (px*M-py*M+2*px*py*T)/(px*M+py*M-2*px*py*T)

    clearly tx=ty only if px=py. If px < py, then x is taxed more than y. Likewise, if pxy, du/dx < du/dy and so demand for x is less elastic than demand for y. The less elastic goods are taxed more.

    However, why don't I get tx=ty? What assumptions have I violated, that must be made in order to get tx=ty? What am I doing wrong here?

  41. 41 41 Mike H

    Erratum : for “Likewise, if pxy” read “Likewise, if px is less than py”

    I blame HTML.

  42. 42 42 Mike H

    Sorry, it should be “if px is less than py, then x is more than y, and du/dx is less than du/dy”

  43. 43 43 David

    @Mike H

    Your consumer choices are odd, i would think it should be

    X=M/(2*(Px(1+tx))) and similarly for y…I’d that / a type on your part?

  44. 44 44 David

    Addendum,

    Ok it’s early here, maybe that is what you meant to write, sorry.

  45. 45 45 JohnE

    @Mike H

    px*tx*x+py*ty*y=T

  46. 46 46 Marek

    @ Mike H: your consumer choices are right, but your budget constraint for the government is wrong. It should be px*tx*x+py*ty*y=T. Then you get tx=ty.
    This example is not very revealing, however, because both goods have the same elasticities, and both prescriptions hold simultaneously.

  47. 47 47 Marek

    I now see that JohnE already has pointed this out, my apology for duplicating the point.

  48. 48 48 JohnE

    @Mike H

    Try U(x,y)=x+ln(y). Then elasticity of demand is not equal for the two goods, yet you still get tx=ty.

  49. 49 49 Mike H

    Ok, thanks. I suspected it was something simple.

  50. 50 50 Mike H

    Ok, now I’m getting tx=ty=T/(M-T) for a whole lot of different u(x,y), independent of px and py. So I can derive (B), no doubt for arbitrary u(x,y).

    A bit of googling suggests that taxing inelastic goods is the result you get when you want to minimise deadweight loss, rather than maximising utility. I must try this next and see what answers I get.

  51. 51 51 Harold

    Lets see if I have even a slight handle on this. For elastic goods, the indifference curves show that equal taxes lead to the most preferred solution for the same revenue. For inelastic goods, the deadweight loss is reduced. A quick example. If we tax wine, then less wine will be produced, everyone is a bit worse off. If we tax heads, there will be no reduction in heads, and no deadweight loss.

    When the indifference curves are produced, do they assume there is no deadweight loss? Would the curves look a bit different if we included this?

  52. 52 52 JohnE

    Here are my thoughts:

    The first conclusion is derived from general equilibrium analysis. The second is derived from partial equilibrium analysis.

    Steve demonstrates what general equilibrium analysis is with his indifference curve diagram. The key point is that general equilibrium analysis shows how the demand for one good changes when the price of a second good changes.

    Partial equilibrium analysis ignores these affects. Thus when considering a tax on an inelastically demanded good, the affect on demand of a second (relatively elastically demanded) good is ignored. This makes things simpler, but it can lead us to the wrong conclusions sometimes, as shown here. Thus the first conclusion is correct and the second is wrong.

    To illustrate, consider Harold’s statement: “If we tax heads, there will be no reduction in heads, and no deadweight loss.” This is wrong in a general equilibrium analysis since taxing heads leads to less income, which means the demand for wine shifts to the left. This leads to a deadweight loss in the wine market.

    Also to clarify, in a general equilibrium analysis, minimizing deadweight loss and maximizing utility are the same thing.

  53. 53 53 Harold

    JohnE: “To illustrate, consider Harold’s statement: “If we tax heads, there will be no reduction in heads, and no deadweight loss.” This is wrong in a general equilibrium analysis since taxing heads leads to less income, which means the demand for wine shifts to the left. This leads to a deadweight loss in the wine market.”

    I had figured that in a two component system, it made no difference where the tax went, the perfectly inelastic good or the other. The amount spent on each is the same in any case. Why is it that the inelastic good should be taxed more heavily?

  54. 54 54 Joel

    Why don’t we take a utilitarian approach – raise taxes is such a way that maximizes economic benefit, to consumers, sellers, and recipients of tax resources.

  55. 55 55 nobody.really

    Friends, Romans, blog-readers: I come not to solve the puzzle, but to quibble about it.

    Landsburg asks us to reconcile (paraphrasing) the goal of taxing all goods equally with the goal of taxing inelastic goods heavily. To my mind, these are both strategies for collecting revenues while managing “dead weight social loss.” But one strategy is superior to the other, at least in the stylized world of econ models. So I don’t see to point of reconciling them; I would apply one strategy until I couldn’t do so any further, and then apply the other. Problem solved.

    Here’s my take: Classical economics argues that if government services could be provided at no cost, prices would move to marginal cost, giving people signals to organize their affairs in a socially optimal manner.

    Taxation produces various effects relative to this hypothetical untaxed world. First, it finances government services. Second, it leaves people with less income than they had in the hypothetical untaxed world (the “income effect”). Third, it (often) causes prices to differ from marginal cost. These latter two effects will cause people to alter their behavior relative to the hypothetical untaxed world.

    I can’t envision how to have taxes and avoid the income effect. But I can still strive to design taxes to minimize distortions to people’s behavior arising from distortions in marginal cost.

    Ramsey observed that the extent to which marginal cost influences people’s behavior depends upon people’s elasticity of demand for any specific good. If my demand for a good is elastic, I’ll buy less of it when the price rises – including when taxes rise. In contrast, if my demand for a good is perfectly inelastic, then marginal cost is irrelevant. I’ll buy what I demand (and only as much as I demand) constrainted only by the limits of my resources. That is, I’m influenced by the income effect, but not by distortions to marginal cost.

    Consequently the most efficient taxes are taxes that avoid distorting the marginal cost of goods – unless the demand for those goods is inelastic. In other words, if you want a tax that minimizes dead-weight social loss, you want a lump-sum tax on something that people can’t evade — such as a head tax.

    Where we can’t or won’t pursue that outcome, the next-best strategy is to tax things in such a manner as to cause all prices to rise proportionately – in effect, emulating inflation. (In practice we have difficulty taxing all goods equally; famously, people evade taxes by substituting untaxed goods – leisure or black market goods and services – for the taxed ones. For purposes of this puzzle, Landsburg has invited us to put these concerns aside.) This tax may avoid substitution distortions, but it still reduces the price signal that would encourage income-generating behavior. Consequently I regard this as an inferior taxation strategy.

    Because I regard taxation of all goods equally to produce inferior results to imposing a lump-sum tax on things people can’t evade, such as goods for which demand is inelastic, I don’t recognize the need to reconcile these two strategies (except for purposes of this puzzle). Instead, I would apply the superior strategy and, if that doesn’t produce sufficient revenue, fall back on the inferior strategy as needed.

  56. 56 56 nobody.really

    Ya know, sometimes it’s helpful to talk these things out. Friends, Romans, blog-readers, I may find myself praising this puzzle yet.

    Where we can’t or won’t pursue that outcome, the next-best strategy is to tax things in such a manner as to cause all prices to rise proportionately – in effect, emulating inflation. (In practice we have difficulty taxing all goods equally; famously, people evade taxes by substituting untaxed goods – leisure or black market goods and services – for the taxed ones. For purposes of this puzzle, Landsburg has invited us to put these concerns aside.) This tax may avoid substitution distortions, but it still reduces the price signal that would encourage income-generating behavior. Consequently I regard this as an inferior taxation strategy.

    Let me walk that one back.

    What exactly is the harm in “reduc[ing] the price signal that would encourage income-generating behavior”? People have 24 hrs/day to invest in some activity. They’ve got to invest it somewhere. And, under the assumptions of this puzzle, the returns on all substitute investments (including leisure) have been similarly reduced. Thus, the fact that a tax induces a change in marginal prices would not cause any change in behavior (other than changes caused by the income effect).

    Ok, so now we have two strategies for raising revenues that would not distort behavior relative to a no-tax world, except to the extent caused by the income effect. With each form of taxation, people are left with no viable substitutes that would enable them to increase their utility – either because the utility they derive from a good/service is sufficiently large as to render the added tax irrelevant, or because the utility of any substitute good/service is sufficiently reduced as to render the choice unappealing.

    Perhaps there’s some way to phrase this insight so as to form an answer to the puzzle.

    p.s. Within this stylized world, I see only one distinction between these taxes. If government recovers a disproportionate share of taxes from goods for which people have inelastic demand, if the price of these goods is discrete/lumpy, and if people stop consuming these goods due to lack of income – that is, they literally cannot afford it – then government would lose all its income. In contrast, if government taxes all goods proportionately, government should continue collecting revenues so long as anyone has at least SOME income.

  57. 57 57 Marek

    One paper to look at is “The Structure of Indirect Taxation and Economic Efficiency” by Atkinson and Stiglitz (1972). Both rules are special cases of a more general principle.

    The second rule is usually derived in a partial equilibrium setup (or when there are no income effects).

    One situation when the first rule for applies is when there is a factor that is supplied inelastically. Then tax on this factor is in effect a lump-sum tax, is nondistorting, and is optimal. Equivalently, a UNIFORM tax on all OTHER commodities is nondistorting. This is what’s going on in Steve’s blue plan/red plan example. The budeget constraint is PX*X+PY*Y=M and M is a fixed factor. Taxing M, equivalently, taxing X and Y uniformly, is optimal.

    If we changed the example and allowed for endogenous labor then it would, in general, no longer be optimal to tax both X and Y uniformly. (unless the preferences are homothetic, which is a second case when a uniform tax on all commodities is still optimal).

  58. 58 58 Harold

    Another go. We have an inelastic thing – say heads, and a basket of elastic things. If we put all the tax on the inelastic thing, this is the same as taxing all the elastic things equally. The tax spreads itself out effectively accross the other goods in terms of consumption. As Marek said, the lump sum tax is effectively the same as a uniform tax on all OTHER goods, so either is non distorting.

    The result is that whatever we tax the inelastic thing, it spreads itself out over the other goods. Therefore it is the same as a uniform tax.

  59. 59 59 nobody.really

    If we put all the tax on the inelastic thing, this is the same as taxing all the elastic things equally. The tax spreads itself out effectively across the other goods in terms of consumption.

    Note that the two taxes seem equivalent within the assumptions of the puzzle. Under less constrained (more realistic) assumptions, they wouldn’t be.

    I can’t envision how we could implement a tax that actually applies uniformly to everything. For example, taxes generally do not apply to leisure or black market goods. And taxes apply non-uniformly to the utility I derive from using goods I own, or exploiting my own labor. If I buy a house and rent it to you, the rent you pay to me is subject to tax; if I buy a house and rent it to me – that is, I live in it – the rent I (impliedly) pay to myself is not subject to tax.

    Only when you appreciate the impossibility to taxing everything uniformly can you appreciate the rare efficiency of taxing inelastic things. And that efficiency derives precisely from the fact that it is NOT equivalent to spreading taxes across all taxable things. A lump sum tax does not distort price signals. In contrast, a tax spread over all taxable things would distort price signals because, as noted above, not all things are taxable.

    (Of course, lump sum taxes also have drawbacks; in particular, they tend to be regressive and hard to administer, in that they tend to be unrelated to ability to pay.)

  60. 60 60 andy

    “Minimize distortions” means that taxing somebody should make him ideally give up on least-valued-goods only with no change in the rest?

    Taxing inealastic goods is roughly the same as ‘taxing heads’; if it is inelastic, it doesn’t matter what % I tax. Taxing % elastic goods has bigger deadweight-loss.

    The more inelastic the good is, the less distortion the tax on this good has. The more elastic a good is, the more dead-weight loss there is, but if you happen to tax substitutes differently, there will be distortion as to consumer switching patterns of consumption.

    Tax inelastic goods more heavily; if you happen to tax elastic goods, tax it uniformly, you will not avoid dead-weight loss, but at least you avoid consumption-pattern-change-distortion.

    (it seems to me that if you draw indifference curves with one elastic and one inelastic good, any tax on anything will ‘minimize distortion’ – the consumer will always reduce the consumption of the elastic goods no matter what you tax)

  61. 61 61 James Knight

    Don’t know if this MP has got all his facts right, but he might have, and it might be a reason to be indignant!

    The annual Sunday Times Rich List yields four very important conclusions for the governance of Britain (Report, Weekend, 28 April). It shows that the richest 1,000 persons, just 0.003% of the adult population, increased their wealth over the last three years by £155bn. That is enough f…or themselves alone to pay off the entire current UK budget deficit and still leave them with £30bn to spare.

    Second, this mega-rich elite, containing many of the bankers and hedge fund and private equity operators who caused the financial crash in the first place, have not been made subject to any tax payback whatever commensurate to their gains. Some 77% of the budget deficit is being recouped by public expenditure cuts and benefit cuts, and only 23% is being repaid by tax increases. More than half of the tax increases is accounted for by the VAT rise which hits the poorest hardest. None of the tax increases is specifically aimed at the super-rich.

    Third, despite the biggest slump for nearly a century, these 1,000 richest are now sitting on wealth greater even than at the height of the boom just before the crash. Their wealth now amounts to £414bn, equivalent to more than a third of Britain’s entire GDP. They include 77 billionaires and 23 others, each possessing more than £750m.

    The increase in wealth of this richest 1,000 has been £315bn over the last 15 years. If they were charged capital gains tax on this at the current 28% rate, it would yield £88bn, enough to pay off 70% of the entire deficit.

    It’s enough to pee us off, isn’t it Mr. Landsburg?

  62. 62 62 Mike H

    Hum… so I find that minimising deadweight loss is NOT equivalent to maximising utility, for the case of a since taxpayer with total money $10 and utility x+log(y) for consumption x and y of good with unit price $1, when we want to raise $1 of tax.

    utility is maximised when the goods are taxed at 11.11% each. To minimise deadweight loss, I tax x at 10.48% and y at 16.44%.

    In the first case, utility is 8, deadweight loss is 0.1111.
    In the second case, utility is 7.9986, deadweight loss is 0.1078.

    So, why would people try to minimise deadweight loss?

    Am I missing something here? Is the thing I’m missing just a mistake in my algebra? I’m defining deadweight loss as the difference between the price of the consumed goods and the price of the goods that would be consumed if there was no tax, which I think is correct under the assumptions of the puzzle. Am I wrong?

  63. 63 63 Marek

    Mike H: Deadweight loss is not a difference between two prices. It is, in this case, the difference between the loss of consumer surplus and the amount of taxes collected (that little triangle under the demand curve..)

  64. 64 64 iceman

    nobody.really, Harold and all others (laypeople in particular) who teased out the idea that taxing inelastic goods is effectively (theoretically) equivalent to taxing all elastic things equally – thanks for that clever insight, hoping you’re on the right track. Maybe we’ll get an ‘official’ answer soon?

  65. 65 65 Greg W

    If the goal is to ensure that all relative prices are unchanged by taxation (i.e. we’re trying to minimize distortion), then we’d just have to tax leisure.

    The best way I can think of to tax leisure is to give workers a tax refund for not taking days off of work.

  1. 1 Tuesday Solution at Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics
  2. 2 Thursday Solution at Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics
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