Thursday Solution

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.

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39 Responses to “Thursday Solution”


  1. 1 1 Andy

    I feel duped… I got the impression that the two statements were to be taken as absolute truths, but I guess this was just a very suttle way to make the point that many economics courses are a bit crap ;)

  2. 2 2 Bill Bethard

    As an aside, Frank Ramsey was also the first Sixth Man in basketball and highly esteemed by his Cambridge-Boston Circle colleagues B(ertrand?) Russell and A. “R.” Auerbach.

  3. 3 3 Harold

    I am not sure if I have this, or my understanding is too simplistic.

    The simple version of the Ramsey rule – tax inelastic goods more – I am OK with. This reduces deadweight loss because the little triangle in the supply / demand graph is smaller. The full version should include the effect on other goods – the cross elasticity – i.e. it should account for the increase in size of all the other little triangles. That seems fine.

    If labor is inelastic, income is effectively fixed. If prices rise, people will not for whatever reason just work more to raise income to compensate.

    When we include cross-price elasticities, we are in the root-beer and peanuts situation. Even perfectly inelastic goods will have high cross-elasticity; taxing them reduces consumption of other things proportionately. Equal taxation is optimum.

    If labor were elastic, then we reduce the cross-elasticity with other taxed goods. In response to tax people have a choice. They can reduce leisure (increase income) and maintain consumption of the elastic goods, or reduce consumption and maintain leisure. With tax on the inelastic good only, if they choose to reduce leisure and maintain consumption, then there are no little triangles for the elastic goods as there is no tax on them. However, if we tax everything equally, then when they trade off between leisure and elastic goods, they are introducing the deadweight loss for the elastic goods. So taxing inelastic goods allows a trade off between leisure and goods without introducing distortions.

    This is why taxing inelastic goods is best if everything cannot be taxed. Or is it?

  4. 4 4 Ken B

    Steve: Since you hit me with Euler-Lagrange before my morning coffee I feel justified in being a bit obtuse here. So some questions and complaints.

    1. When M is square it is invertible iff it has nullity 0. You assert M has co-rank 1 but invert it. So what am I missing?

    2. I think you should define Uj

    3. What individual maximization? Ditto budget constraint.

    4. You maximize sum Ti*Xi subject to U constant. From the puzzle I’d say the more natural understanding of “what’s the best tax mix” is maximizing U given a fixed net tax revenue. How to make people best off given we need to raise N dollars. Am I missing an equivalence?

  5. 5 5 Steve Landsburg

    Ken B: I don’t know how that “corank 1″ comment slipped in there; I’ve deleted it.

    As for 4, yes, you are missing an equivalence.

    Individuals maximize U(X1,…Xn) subject to the constraint sum(PiXi) = Income, which is taken to be fixed.

  6. 6 6 martin henner

    The analysis given seems to imply that all funds will be spent on goods. Whatever is left over after rootbeer is spent on peanuts.

    But what if that is not the case, and the changes in price just leads to more or less money put into savings, rather than purchases.

  7. 7 7 nobody.really

    Yea! Screw immigration; let’s talk public finance.

    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”.

    I’m not following this.

    Does it make sense to talk about cross-elasticities of demand for things with inelastic demand? Imagine that peanut butter consumption has cross-elasticity with both jelly consumption (positively correlated) and baloney consumption (negatively correlated). As any parent knows, demand for peanut butter is inelastic. So if we tax peanut butter, consumption of it does not change (except due to income effects – that is, people simply cannot afford it anymore). Thus the tax would also not affect consumption of the things with which it has cross-elasticities (again, except for the result of the income effect).

    So what is the consequence of taxing things with inelastic demand on things with ELASTIC demand? The demand for left shoes is cross-elastic with the demand for right shoes. If we increase the tax on peanut butter, the only effect on the demand for left and right shoes I can anticipate is the income effect.

    Can anyone think of an example in which we increase the tax on something with an inelastic demand, and it leads to some distortion (other than a distortion related to the income effect)?

    If you can’t tax leisure (or equivalently subsidize labor) then it’s not in general optimal to tax everything else equally.

    Is it possible to avoid the distortions of taxation by subsidizing (paid) labor to keep it as attractive as leisure, while also taxing (sold) goods? Again I confront the idea that we each have 24 hrs/day to spend, and that we

    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.

    First, we should acknowledge that the labor/leisure trade-off is not the only challenge in the effort to tax everything equally.

    But more importantly, I wonder if we’re again wandering into the “so misleadingly that it might as well be wrong” territory when we suggest that the supply of labor is inelastic. I understand that phrase to mean that we can’t produce skills instantaneously to respond to increased demand. But this phrase does not refer to the supply of unskilled labor. Nor to the idea that, at that margin, taxes might discourage laborers (including skilled laborers) from working.

    Believe me, I’d be only too happy to argue that we can tax labor without consequence to the labor supply, if only I could find support for that idea.

  8. 8 8 Ken B

    @Steve:
    re 4 “Individuals maximize U(X1,…Xn) subject to the constraint sum(PiXi) = Income, which is taken to be fixed.”

    I still don’t see your equivalence. I am saying “it looks like you maximize tax revenue given a fixed value for U, but isn’t that backwards, aren’t we maximizing U given a fixed level of taxation?” To belabor:
    It looks to me like you are maximizing the wrong thing. Your latest comment suggests your contraint problme is “U and sum Pi*Xi fixed, find max Ti*Xi” Mine is “sum Pi*Xi fixed and sum Ti*Xi fixed, find max U”
    So why is your problem equivalent, or if different, the correct one?

  9. 9 9 Neil

    The most general form of the Ramsey rule, and the one I like the best, is that the price effects of taxes (that is, the compensated effects) should result in an equiproportional reduction in the consumption of all taxed goods. In the case where the demands for taxed goods are independent this implies the inverse elasticity rule if leisure is not taxable and not fixed. But this statement of the Ramsey rule is more general because it holds when taxed goods are not independent as well. To me, it also expresses a more general form of tax neutrality. We know that tax systems reduce consumptions of taxed goods, and the general Ramsey rule says this reduction should happen uniformly across all taxed goods. If leisure is taxable or fixed, this uniform reduction can be accomplished by uniform tax rates. But it is the uniform effects of taxation on consumptions that is the general meaning of tax neutrality.

  10. 10 10 nobody.really

    We know that tax systems reduce consumptions of taxed goods, and the general Ramsey rule says this reduction should happen uniformly across all taxed goods. If leisure is taxable or fixed, this uniform reduction can be accomplished by uniform tax rates. But it is the uniform effects of taxation on consumptions that is the general meaning of tax neutrality.

    To the extent that I understand this, I disagree with it.

    As far as I can tell, the conceptually ideal tax is the lump-sum tax. This tax, as any tax, makes people poorer, but does not alter marginal cost price signals.

    Now, how do I respond to price signals? That depends on my preferences and resources. If I’m rich, I may respond by eating steak; if I’m poor, I may substitute hamburger.

    Compared to a hypothetical world in which I receive government services for free, I’m poorer in a world in which I must pay for government services. Would an efficient tax system prompt me to maintain the consumption patterns of a rich person, but reduced on a uniform basis to fit my new income (e.g., continuing to eat steak, but smaller steaks)? Or would it simply leave me with a lower income, and let me decide how to modify my consumption patterns to compensate (e.g., substituting hamburger for steak)? I propose the latter.

  11. 11 11 Neil

    nobody.really

    The optimal tax problem presumes lump-sum head taxes are not possible, and that taxes have to be levied on economic activities. Why else would we be asking about how tax rates should be set?

  12. 12 12 Steve Landsburg

    Martin henner — think of savings as just another good

  13. 13 13 nobody.really

    The optimal tax problem presumes lump-sum head taxes are not possible, and that taxes have to be levied on economic activities.

    I think of taxation on goods for which the demand is inelastic as the equivalent of a lump-sum tax, in that it extracts funds but does not distort marginal cost.

  14. 14 14 Neil

    If there were a perfectly inelastic good, that is right and you’d get all your revenue from taxing it alone. The Ramsey problem applies to the case where goods have variable degrees of elasticity, not just the case where there is a good that is perfectly inelastic and the equivalent of a lump-sum tax is possible. In short, the Ramsey problem asks “how should we tax when lump-sum taxes are not possible?”

  15. 15 15 nobody.really

    In short, the Ramsey problem asks “how should we tax when lump-sum taxes are not possible?”

    Great; that’s a more interesting question anyway.

    My main point was to question whether we should structure taxes in a way to defeat a person’s natural tendency to substitute inferior goods for superior goods as taxes increase. Yes, an efficient tax will minimize the “distortion” in people’s behavior caused by changes in marginal price signals. I’m not persuaded that substitutions due to wealth effects count as a distortion.

  16. 16 16 Mike H

    @KenB

    If I minimise x+y subject to fixing xy=1, I get x=y=1 (assuming x,y>0)

    If I maximise xy subject to x+y=2, I get x=y=1. Same solution. The problems are equivalent.

    In general, any problem of the form “optimise U subject to keeping T fixed” is equivalent to a problem of the form “optimise T subject to keeping U fixed” – the solutions are the same.

    Or think about it this way. Suppose things aren’t optimal. The tax department could look, and say “hey, we could change the tax rates, get more money, without hurting anyone!”, likewise, the voter could say “hey, the government could change the tax rates, still have the same money, and we’d all be better off!”

    If things are optimal, the tax department says “ok, if we want more money, we can’t do it by shuffling tax rates without making the voters unhappy” and the voters say “ok, if we want better tax rates, we can’t do it without worsening the deficit”

    The optimal solution for maximising U subject to T is the same as the optimal solution for maximising T subject to U.

  17. 17 17 Mike H

    @Steve

    thanks for posting this puzzle and the solutions (technical and non-)

    I learned something new this week :-)

  18. 18 18 Steve Landsburg

    Neil: Your comment starting “The most general form of the Ramsey rule….” is terrific. Thanks for it.

  19. 19 19 Ken B

    @Mike H: Maximize the perimeter holding the area fixed. Cannot be done. Maximize the area holding the perimeter fixed: circle.

    You can give the first problem a solution by adding constraints like convex shape and a maximum diameter, but the problems still don’t give the same answer.

  20. 20 20 Mike H

    @Ken B

    no, but “Minimize the perimeter holding the area fixed.” can be done, and gives a circle.

    For f=x+y and g=xy, at the optimum, for infinitesimal changes in x and y, as f goes up, g also goes up. Hence, minimisation on g is the same as maximisation on f and vice-versa.

    For areas and perimeters, at the optimum, for infinitesimale changes in shape, as A goes up, P also goes up. Hence, minimisation of perimeters is the same as maximisation of areas, and vice-versa.

    For tax take and utility, at the optimum, for infinitesimal changes in the tax rates, as T goes up, U goes down. Hence, maximisation of T is the same as maximisation of U.

    If you want to analogise this problem into areas and perimeters, then maximisation of U (keeping -T fixed) is the same as minimisation of -T (keeping U fixed).

    Or, if you prefer, write down the lagrangian for the two problems. On the one hand, you have T+aU. On the other, you have U+bT. Letting a=1/b, you see that the one problem is eqivalent to the other, unless a or b is 0.

  21. 21 21 Al V.

    So, thinking about real world application, perhaps the most inelastically demanded resource is (drum roll) health care. Is the implication that we should tax health care/health insurance at the same rate as other products?

    If savings are just another good, then shouldn’t we tax savings at the same rate? Doesn’t that imply that capital gains should be taxed at the same rate as other income? From the perspective of a person making an investment, the investment is just a form of savings – a purchased good, where the good I am purchasing is the return on my investment.

  22. 22 22 iceman

    Al V – I expect the answer on savings would be yes, absent an income tax. It seems an income tax is a sort of tax on all goods (whether optimal or not), including savings, and an additional tax on the return to savings distorts the intertermporal preferences.

  23. 23 23 vik

    ‘When all goods can be taxed, that complicated formula does in fact tell you to tax them all equally’- yes if income effects don’t swamp substitution effects (this would show up in cross-elasticities) and if all agents can compute and move frictionlessly from one permanent income based consumption profile to another and if agents can’t form a coalition to redistribute tax burdens and so on.
    In any case ‘goods’ can’t be well defined. I don’t go to work but spend the day painting my house- this is leisure. I go to work and pay a guy to paint my house- that’s labor. I’m a spendthrift and blow my entire salary on cases of champagne- that’s consumption. I invest my in vintage wine- that’s saving.
    If elasticity of supply are assumed to be perfect it follows that the elasticity of derived demand must be perfect. But all demand is derived demand. If I’m not supplying any factor of production under this regime, chances are I’m no longer here. My demand for stuff was derived demand. I didn’t want bread I wanted whatever bread turns into when I chew on it and swallow it. I don’t want a 3 d tv, I want the qualia I get when I watch the stupid thing.

    Nor can all goods be taxed. Otherwise just tax crime out of existence. Obviously taxes are set on the basis of their cost of collection, their degree of social acceptability and so on.

    I’m not saying a flat V.A.T on every economic activity (including Religion and Prostitution) ain’t a good thing. Nor do I fail to see why people like Joan Robinson & Frank Ramsey would see merit in this result. Equal taxes on goods is the other side of the coin of Equal Rights (i.e. obligations on the Govt.). There is a pernicious sort of ‘fiscal drag’ associated with both which can stoke up a social catastrophe- as appears to be happening in Greece and now Spain- as an alternative to a perhaps equally damaging monetary shock.
    I recall Bhagwati’s story about Joan Robinson. She was heard agreeing with Guy Ranis, then a Professor at Harvard, that Korea really had made remarkable progress. It turned out she meant North Korea.

    The problem here is that

  24. 24 24 vik

    Steve- here’s a question for you- suppose there was no Economics, no Functionalist Sociology, nothing save the notion of mimesis, mirror neurons, Gabriel Tarde’s Laws of imitation.
    Suppose, further, that path dependence, Concurrency and ditopology were where all the Research money was at.
    In other words, imagine a Universe in which preferences were never independent, band-wagon effects ruled everything.
    What would be the Ramsey rule in that Universe?

  25. 25 25 Al V.

    So, if we call savings just another good, and we tax all goods equally, then what is the difference between a consumption tax and an income tax? The only things I can do with my income is spend it or save it.

  26. 26 26 Steve Landsburg

    Al V:

    So, if we call savings just another good, and we tax all goods equally, then what is the difference between a consumption tax and an income tax?

    A tax on labor income is equivalent to a consumption tax whose rate does not vary over time.

    A tax on all income (both labor income and capital income) is equivalent to a consumption tax whose rate rises over time.

  27. 27 27 Mike H

    @Al V.

    On healthcare : there are positive externalities here that take the issue out of the bounds of this puzzle. If you are healthy, others benefit too. Pigovian subsidies of healthcare may therefore be necessary to restore efficiency.

    On savings as a good : Steve Landsburg often distinguishes “Scrooges”, who obtain utility* from their hoards of cash, from everyone else, for which a pile of cash is mainly just a ticket to future consumption, providing not much utility in its own right. For the Scrooges, savings might be a good. For most of us normal everyday ducks, it’s not.

    Footnotes:
    * He loves to dive around in it like a porpoise, and burrow through it like a gopher, and toss it up and let it hit him on the head [1]

    References:
    [1] Barks, C “Only A Poor Man” Uncle Scrooge 1 pp2-33 (1952)

  28. 28 28 Steve Landsburg

    vik:

    ‘When all goods can be taxed, that complicated formula does in fact tell you to tax them all equally’- yes if income effects don’t swamp substitution effects (this would show up in cross-elasticities)….

    This is incorrect. The argument for taxing all goods equally has nothing to do with the relative sizes of income and substitution effeects.

  29. 29 29 John Berkowitz

    @Mike H

    Maybe I don´t get it right, but what you are trying to say is that money as savings is not utile for the average person who does not have enough savings to really make something out of it. I´m not sure if I should agree or disagree, because I don´t have any data about how much money is in the finance markets from average people and how much money is there from the “Scrooges”.

    John McMurtry talks about the Money to more money sequence. Following his theory, the unproductive money (money not invested into producing something real) is the most important part of our capitalist society.

    The other problem I have that if there are 72% Canadians with debt (quoted from Three quarters of Canadians in debt), how does this influence the tax system? I believe that those people can´t really choose between the leisure time and more working, since they are inside their own loose-loose scenario (depending of course on the levels of their debt).

  30. 30 30 vik

    ‘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.’

    I don’t understand this. Ramsey’s rule says tax Giffen type goods (with big income effects- presumably consumed by poor people) at a lower rate.
    Am I missing something?

  31. 31 31 Steve Landsburg

    Vik:

    Am I missing something?

    You most assuredly are.

  32. 32 32 vik

    What precisely?

  33. 33 33 Steve Landsburg

    Vik: You seem to have missed the following points:

    1) This is a question about efficiency, not redistribution.

    2) We are talking, in any event, about an economy with identical consumers.

    3) Lump sum redistribution together with efficient taxation beats inefficient taxation every time BY THE DEFINITION of efficiency.

    Putting 2) and 3) aside, 1) is the key to where you went off the rails.

  34. 34 34 Harold

    I believe that one of the arguments about Ramseys rule is that its application taxes the poor a lot. Some consider it of overall benefit to more progressive, but less efficient tax system. The important thing is to recognise the costs and benefits of each.

  35. 35 35 iceman

    Mike H – “If you are healthy, others benefit too.” A view that sounds reasonable in the micro, but begs troubling questions in the macro:

    1) If I am happy, others benefit too. So what else should we subsidize? What if drinking 17 oz. sodas makes me happy?

    2) If it derives via a self-imposed moral obligation to pay for others’ HC, the ‘externality’ seems contrived (e.g. really reducing the cost of a negative externality that we created).
    If via general benefits of greater productivity,
    a) is this a benefit to which we can really claim to be entitled, e.g. what if I simply prefer to consume more leisure?;
    b) this would seem to have less to do with access to HC services (for those who lack it) than with general lifestyle habits (which are the source of much demand for HC), which brings us back to taxing “sinful” goods — and possibly justifying much broader social engineering.

  36. 36 36 Mike H

    @John, there are arguments to the effect that the mere existence of a big stash of cash (or other savings) provides a stream of benefits that is independent of the benefits obtained by spending it in the future. For example, it provides a hedge against uncertainty – a free insurance policy, if you like. To the extent this is true, savings are a good in their own right.

    @iceman(1) The Mirrlees review of UK taxation recommends against applying Pigovian taxes too widely. Presumably the same principle would apply to Pigovian subsidies too.

    @iceman(2) The govt would probably get much more bang for their healthcare buck if they started taxing fat as well as subsidising statins. I don’t believe that externalities from healthy choices derive solely from an obligation on others to care for the sick, no more that externalities from choosing education derive solely fro, an obligation on others to care for the jobless.

    Note, by the way, that healthcare is subject to a multitude of market failures, not just the existence of externalities. Each one of these enables one to make a case for sensible government intervention.

  37. 37 37 iceman

    Mike H – that’s my point, how do we decide where to draw the line, and based on what externalities (you brought them up). You offer no guidance just a preference.

  38. 38 38 Mike H

    @iceman

    Coase’s work implies you need evidence about the effect of policies before you decide which side you plonk Pigovian “remedies” onto.

    So, let’s look at evidence – how is the US doing, healthwise vs dollars spent, compared to other developed countries?

  39. 39 39 He-man

    Sorry to be joining this thread a little late. Actually this result really bothered me as squaring Ramsey with uniform taxation is no small feat (the two prescriptions are in many senses polar opposites) and the author offers very little intuition for it. However, now that I’ve had a chance to look into it, it turns out that the result is a bit of a cheat. So for those that can be bothered, here’s (my version of) the intuition.

    Firstly, we know that taxes are distortionary in the sense that they alter the relative prices of goods (including those for working and leisure) and therefore they alter the trade-offs that people face when making economic decisions;

    Secondly we know that in theory, lump-sum taxes are optimal as they do not distort prices (as people cannot do anything to reduce their tax liabilities).

    Uniform taxation only emerges as first-best if you really do assume that everything can be taxed; including leisure. To see this note that everyone has a given ‘endowment’ of time, which has a value. This is what we refer to as ‘full income’, and one way or another, all of a person’s full income must be consumed. E.g. it can be converted into monetary income in order to finance consumption on conventional goods and services, or it can be spent on ‘consuming’ leisure (at an implicit price). In other words all of a person’s full-income is spent on ‘goods’ of one sort or another.

    However, if all goods can be taxed then:

    1. taxing all goods at a uniform rate does not distort the relative prices of any pair of goods (i.e. there are no distortions); and

    2. no one can shift the burden of their tax, as all consumption is taxed (effectively each person’s total endowment/full-income is taxed)

    from which it follows that uniform taxation is equivalent to a lump-sum tax as the tax liability cannot be altered.

    So in other words, the manner in which the author manages to square Ramsey with uniform commodity taxation is deceptive (to put it politely); as it’s nothing more than lump-sum taxes in disguise. Of course there are several real justifications for uniform taxation, including:

    - real-world difficulty of gathering data on own-price and cross-price elasticities to implement Ramsey
    - lower administration/compliance costs
    - less scope for lobbying for preferential tax treatment

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