23 Apr 2011

Two Quick Notes on Tax Theory

Economics 11 Comments

[UPDATE: I think I misunderstood what the objective was in David R. Henderson’s post, so you can safely disregard my Note #1 I think. See the first comment by Alex Tabarrok.]

NOTE #1: David R. Henderson repeated an interesting argument against progressive taxation that I had never heard, but I think it’s wrong. David stipulates for the sake of argument that we can make interpersonal utility comparisons, and he even stipulates that there is diminishing marginal utility in money. I.e. David is stipulating (again, just for the sake of argument) that we can say $1 to Bill Gates means less than $1 to Bill Pullman. Even so, David claims these assumptions aren’t enough, to conclude that progressive income taxation increases “social utility” or some such criterion. Here’s David making the argument, and then he quotes Pigou saying it too:

[These assumptions are] consistent with a lower marginal tax rate on the high-income person because even a lower marginal tax rate above some income level will take more money from the high-income person than from the low-income person….[But y]ou need a much stronger assumption, specifically, a very steep decline in the marginal utility of income, to get Johnston’s conclusion [that a progressive income tax imposes “equal sacrifice” among taxpayers].

Here’s how Arthur Pigou put the issue in his 1951 book, A Study in Public Finance:

All that the law of diminishing utility asserts is that the last ₤1 of a ₤1000 income carries less satisfaction than the last ₤1 of a ₤100 income does. From this datum it cannot be inferred that, in order to secure equal sacrifice . . . taxation must be progressive. In order to prove that the principle of equal sacrifice necessarily involves progression we should need to know that the last ₤10 of a ₤1000 income carries less satisfaction than the last ₤1 of a ₤100 income; and this the law of diminishing utility does not assert.

OK I have to confess that I was flummoxed by this for a few minutes, because it seemed Pigou and Johnston were both right. Fortunately, I figured out (I think) what was going on, before my head exploded.

If “progressive income taxation” means that Bill Gates has his entire income taxed at a higher rate than Bill Pullman does, then Pigou is right: We have no reason to suppose we are distributing the utility-pain of taxation equally. (Again, pretend that this language even makes sense, just for this once.)

But that’s not what most people mean by the term. No, a modern “progressive income tax” has various brackets, in which the higher rates apply. So if the last $1 of a $1 million income carries less utility than the last $1 of a $10,000 income, then (giving the progressive all his other assumptions) it makes sense to impose a higher tax rate on income above $1 million, as opposed to that portion of income in the $10,000 – $y,000 bracket.

NOTE #2: Like a stopped clock, Gene Callahan occasionally throws some criticisms against modern libertarian economists that stick to the wall. (Am I mixing metaphors? Perhaps, but I note that most stopped clocks are in fact stuck to the wall.) Here and here he is wondering why we emphasize the distinction between the legal and economic incidence of taxes if and only if it helps our case for lower tax rates. In other words, Callahan is accusing us of doing a Krugman Kontradiction, although in this case it would be a Murphy Muddle.

The worst part is, off the top of my head it’s not obvious how to respond to Gene, and for sure I don’t think any of his commenters have provided an adequate solution. I’m having trouble because when I taught the tax incidence stuff to undergrads, we always used a fixed dollar amount of a tax to make drawing those cute lines (and filling in the deadweight loss triangles with colored chalk!) easier. In contrast, with supply-side griping about income tax rates, we always do it in terms of percentages, so I’m not sure if that has anything to do with Gene’s challenge. Anyway, I recommend bribing Gene’s ISP to disconnect his blog.

11 Responses to “Two Quick Notes on Tax Theory”

  1. Alex Tabarrok says:

    Suppose the tax rate is 10% and consider two people one of whom has $10 of income the other $100. If U(x)=Sqrt[x] then the first person loses Sqrt[10]-Sqrt[9]=.1622 of utility while the second loses Sqrt[100]-Sqrt[90]=.513 of utility so a flat tax has actually imposed more sacrifice on the person with higher income. Thus, just as David said, to impose an equal sacrifice you could well want a *regressive* tax structure. To be precise equal sacrifice means taking no more than 3.2 from the rich guy so any tax structure that leaves him with this will work but if you want to keep 10% on the first 10 at least then the remaining portion has to be regressive.

    Or, you could just assume log utility which means that a 10% tax is equally burdensome on the poor and the rich.

    • bobmurphy says:

      OK I have to think about this some more. I was thinking about equalizing utilities, not equalizing utility losses, and those are obviously different things.

  2. Eli says:

    Regarding #2, Gene writes:

    “Steve Landsburg notes that the economic incidence of a tax is separate from its legal incidence. So, if you try to tax, say, savings accounts of the rich, you can’t really say upon whom the real burden of the tax will fall. I think that’s right.

    But then it can’t also be right to complain that high marginal rates discourage productivity, because that argument assumes the tax burden stays right where one puts it, doesn’t it? Or have I missed something?”

    Unless I’m not understanding the issue, the reason Gene is wrong here should be clear from looking at the S&D graph. Yes, the real burden of the tax will fall proportionately on the more relatively inelastic side, in terms of what % of the tax gets paid by whom on a per unit basis. Let’s say the demand side is more inelastic. Even still, as long as demand is not perfectly inelastic, the firm’s price increase due to the tax will reduce the quantity demanded and thereby discourage production. Another side to the same coin is that as long as demand is not perfectly inelastic, the producer won’t be able to raise his price enough to completely cover the cost of the tax.

    As for Gene’s second blog post, I think he makes a valid point. While I think it probably is true that the wealthy disproportionately fund the US government, it’s not something we can demonstrate a priori.

    • Gene Callahan says:

      Well, Eli, it’s a matter of degree of elasticity and where the ripple effects stop, right? Because there is no reason it has to stop with the demander of the product taxed: Consider a product with a perfectly inelastic demand curve, say, “imaginary cigarettes,” which are even more addictive than the real ones are. In that case, the smokers simply cut there consumption of some OTHER product to keep smoking the same number of cigarettes, so a chunk of the tax falls on that party, and none of the tobacco producer.

      But my larger point was that, to whatever extent we say “Can’t really tell who that tax falls upon,” to that extent it becomes more and more difficult to say whether the tax really represents a high marginal tax rate or not. So sure, in your example the quantity demanded of the firm’s product drops, but it drops less than the purported marginal rate. So what is the REAL marginal rate? To the extent we can’t say where the tax really falls, we can’t say what that rate is.

      • Tel says:

        I disagree, if you crank the tax on cigarettes then you are actually taxing the vendor not the buyer. My logic is based on what level you can continue to crank the tax before you hit a problem. Eventually, people either illegally grow their own tobacco or they buy black market tobacco and bypass the government completely. When this happens, the value proposition for government protection of a controlled market becomes less valuable than the risk involved in bypassing government protection. It is the vendor who wants the protection of a controlled market, not the end-user who really doesn’t care what he/she smokes. Of course, there is a gradual transition since black market tobacco always exists in greater or lesser quantity.

        Well, OK, the end user may also care just a little…

        • Gene Callahan says:

          Tel says:

          I disagree, if you crank the tax on cigarettes then you are actually taxing the vendor not the buyer.”

          So you have no idea what “price elasticity of demand” means.

          • Tel says:

            It means some guy is staring hard at one particular dimension, while the system as a whole moves through all of the degrees of freedom open to it.

            I’m sure your theory does a great job at explaining why alcohol prices reached infinity during prohibition (after all demand continued unabated, but supply was exactly zero by government decree). Great job… if we needed theories to explain events that never happened.

      • Eli says:

        But my larger point was that, to whatever extent we say “Can’t really tell who that tax falls upon,” to that extent it becomes more and more difficult to say whether the tax really represents a high marginal tax rate or not.

        Sure, it becomes harder to tell who bears the major burden of the tax, but we know the magnitude of it and that it will fall somewhere and discourage production in some way, even if not directly for the product in question, as you point out in your imaginary cigarettes example. I don’t see how this prevents us from saying what the marginal tax rate is, or that higher taxes discourage production.

        • Gene Callahan says:

          “Sure, it becomes harder to tell who bears the major burden of the tax…”

          Right. Then you no longer know what the marginal rate of the economic incidence of the tax is. You only know the marginal rate of the legal incidence.

  3. Tel says:

    If I may be so bold, to put government into an Austrian context… government is a buyer and seller at the marketplace just like any other. Government sells services, well actually government sells only one service which is institutionalized violence (or to be more precise you buy the opportunity for favorable treatment in the face of institutionalized violence).

    The polite way to say this is “monopoly on the use of force” and the common man’s way to say it is “protection racket” but either way, it is in fact a service, and it is a service that you want. It is a rather special service because it tends towards a natural monopoly, but not entirely towards a monopoly (there is no world government yet). Thus government comes to the marketplace just like everyone else, and Joe the fruit farmer says, “Who wants to buy my fruit?” but government says, “Ya need protection!”

    Now, back to the question at hand. The rich have more to lose should the waste product hit the rotational air propulsion mechanism. Thus, the rich have more demand for the product and thus they pay more. A poor man dealing drugs in a bad neighborhood might bribe a cop here and there but basically he has nothing to gain by paying tax. Thus, we can say that there is a natural rate of tax based on what people have, what they can afford to pay and what they stand to lose.

    I’ll also point out that if government offers high quality judicial rulings, this actually improves the value proposition of the protection service, so it’s worth paying more tax for a non-corrupt government than for a corrupt government. In the converse, if people who widely believed their government was of a high quality come to discover that it is more corrupt than they previously believed then the new information reduces the value proposition of paying tax.

    Feel free to expand on this basis, or mangle it around to suit the argument wherever necessary,

  4. Tel says:

    Anyway, I recommend bribing Gene’s ISP to disconnect his blog.

    An excellent idea if it wasn’t for the competitive free market. I’d threaten to host it myself if that also meant getting bribes from you not to host it. I love getting paid for not working.