The McArdle Chronicles redux: How to Argue in Bad Faith: An Example

Again, I’m the messenger.  My MIT colleague Jim Bales has taken up my slack in covering the gift that keeps on giving, Megan McArdle.

Enjoy — it’s a good one


Jim Bales here – my thanks to Tom for letting me borrow his soap box. The words that follow are mine, and not his.

So, Megan McArdle has a post in which she asserts that:

“[T]he federal income tax is now very progressive; it collects most of its revenue from people at the top.”

Commenter mmh53b noted:

“When I was a lad … progressivity was not based on the percentage of tax revenue collected from the top, but rather the marginal tax rate.”

The lesson on arguing in bad faith can be found in Ms McArdle’s reply:

‘In this context, the question is: how dependent are tax revenues on high incomes? Because the more dependent they are on high incomes, the more they swing from peak to trough. This has, contra your belief, always been a definition that characterizes a system as “progressive” rather than “regressive”.’

Wow – sucks to be commenter mmh53b, doesn’t it? After all, what mmh53b had always considered to be the definition of a progressive tax is now simply their belief, a belief contra-ed by Ms McArdle with a definition. In fact, Ms McArdle insists that her definition has always been a definition, and thus it is the only definition she will allow for the term “progressive” tax.

Does anyone support poor mmh53b? No one of any importance. Just:

Wikipedia: A progressive tax is a tax by which the tax rate increases as the taxable base amount increases

The Mirriam-Webster Dictionary: Progressive, increasing in rate as the base increases, with the example, “a progressive tax” (definition 4b)

The Encyclopaedia Britannica: [P]rogressive tax, tax that imposes a larger burden (relative to resources) on those who are richer

The Oxford English Dictionary: The only definition to use the word “tax” (2d), reads: Of a tax or taxation: increasing gradually according to ability to pay; increasing as a proportion of the sum taxed as that sum increases. (As to Ms McArdle’s “always”, the first usage cited was in 1792 by Tom Paine.)

Well, it appears that the “a definition” that Ms McArdle prefers is sufficiently rare so as to have been overlooked by the both Britannica and the OED. It certainly seems that “progressive tax” has always meant what commenter mmh53b has thought it meant. Perhaps Ms McArdle, in her capacity as Business and Economics Editor of the Atlantic is using in it a narrow, technical sense? Perhaps economists never use “progressive tax” in its common (and well-nigh eternal) meaning of higher tax rates on higher incomes?

Thanks to Google Books we can quickly check a few Economics textbooks.

The Shrill One (Dr. Robin Wells) and her spouse (some fellow named Krugman) write in their tome Macroeconomics (p. 192) An individual in a higher income bracket pays a higher income tax rate in a progressive tax system like ours. Then again, they are shrill. What do they know?

Now, Robert Samuelson, he was an economist! He’ll get this one right! In his Economics he wrote (page 390, caption to Figure 16-4) Taxes are progressive if they take a larger fraction of income as income rises. Well, maybe Samuelson wasn’t such a good economist, since he didn’t know the “a definition” that Ms McArdle insists was “always” in place.

I know — Professor (and chairman of President Bush’s Council of Economic Advisors) Greg Mankiw will get it right! In his Principle of Economics Mankiw defines a Progressive Tax (p. 255) as a tax for which high-income taxpayers pay a larger fraction of their income than do low-income taxpayers. Oops.

And so we are left with three simple choices.

1)     All of these people, from Tom Paine through Robert Samuelson all the way to Greg Mankiw, are wrong and Ms McArdle is right.

2)     Ms McArdle, the Business and Economics Editor for The Atlantic, is utterly ignorant of the meaning of an economic concept as basic as a progressive tax.

3)     Ms McArdle will make shit up rather than acknowledge that one of her critics was right.

My money is on 3), hence the title of this post. Why? Because her evasiveness was utterly unnecessary. She need only have said, “Why yes, mmh35b, you are correct. A progressive tax system has higher tax rates on higher incomes. As a result, the tax revenues come disproportionately from the wealthy, whose income is more volatile than the less wealthy. Furthermore, that volatility causes tax revenues to go down when the economy tanks, which is when governments have increased need for that revenue.” Of course, had she done so she would have acknowledged that she was sloppy in her choice of words in her original post, and that her critic had caught her out. So, rather than admit the small error and turn it into a chance to advance her cause, she chose to try to shut down mmh35b instead. And that is arguing in bad faith.

PS – A Counter Example

In contrast to Ms McArdle, consider the actions of Mr. Louis Martinelli (ht Abi Southerland at Making Light). After working for many years with the National Organization for Marriage to deny same-sex couples the right to civil unions (much less full marriage), Mr. Martinelli has come out in support of full marriage equality and issued an retraction of his past words and deeds that he now considers to be wrong. In particular, notice in the latter link how Mr. Martinelli holds fast to those elements of his prior statements that he still considers true, yet acknowledges and retracts those elements that were false, irrelevant, or simply hurtful.

One need not agree with Mr. Martinelli completely to recognize that he is striving to argue in good faith. One need not disagree with Ms McArdle completely to recognize that arguing in good faith is not important to her.

Image: Jan Massys, At the Tax Collector, 1539

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25 Comments on “The McArdle Chronicles redux: How to Argue in Bad Faith: An Example”

  1. Jim Bales Says:

    My thanks agin to Tom for posting my thoughts!

    Let me flag the two typos I see now that it is live:

    1) I should, of course, have written “well-*nigh* eternal”, not “well-*neigh* eternal”

    2) In the last sentence before the PS, I omitted the word “advance”. The text should read: “So, rather than admit the small error and turn it into a chance to *advance* her cause, …”


  2. […] another thing:  if you’ve read this far, you might want to check out a much shorter and quite lovely take down of another McArdle folly by James Bales, who blogs with me at my personal site, The Inverse Square […]

  3. Ian Preston Says:

    Neither McArdle nor mmh53b are using the standard definition. A progressive tax, as all of your economic authorities say, is, simply put, a tax with a rising average rate. Of course, that has implications for both marginal rates and for the share of tax paid by the rich but you wouldn’t usually use either of these features alone to define progression.
    A progressive tax has a marginal rate which is above the average rate at all incomes. Comparing the marginal rate with the average rate can therefore be a way of defining and measuring progression. Just looking for a rising marginal rate is not a sensible approach though. It is easy to construct a tax schedule with a rising marginal rate everywhere which nonetheless redistributes in relative terms from poor to rich. The definition proposed by mmh53b is not only therefore not the definition, it is not even a good one.
    Under a progressive tax the rich contribute a high share of tax revenue relative to their share of income. Comparing their share of revenue with their share of income can therefore be a way of defining and measuring progression. Just looking at whether the rich pay a big share is not sensible though. The rich pay a greater share so long as marginal tax rates are positive, do so to a greater degree the more unequal pretax incomes are, however progressive the schedule is and even if the tax system is redistributing in relative terms from poor to rich . The definition proposed by McArdle is not therefore good either.
    What she should therefore have said is more like: ” “Why, mmh53b, we are neither of us being very precise …”

    • Bob Says:

      Here’s a hint though: McArdle would never ever do that because she can’t admit being wrong without mentally throwing a hissy fit.

      Btw, there are a lot of us who have read her on and off for years and just are mindboggled at how poor her writing is, but it’s a feature not a bug at this point. This is the sort of stuff people would write in high school and in college papers to try to bullshit their way past professors.

    • Jim Bales Says:


      Many thanks for the correction! If I understand correctly, mmh53b’s definition (and the one I mistakenly identified as equivalent to those definitions I cited) is that the *marginal* tax rate be monotonically increasing (or at least non-decreasing), while the actual definition requires that the *average* tax rate be monotonically increasing (or at least non-decreasing).

      If so, it seems to me (i.e., I haven’t proved it to the satisfaction of a mathematician) that mmh53b’s definition captures a subset of all progressive taxes. I cannot conceive of a function f(x) which is both:
      a) Monotonically non-decreasing, and
      b) Has the property that {the definite integral [x*f(x)] divided by the upper limit of the integration} can ever be decreasing.

      If any of you (not just Ian) can give me a counter-example, I’d love to see it!

      [Caveat — the above ignores important bits of the real world like deductions, exemptions, and tax credits. Ian, if your comment incorporates any such real elements of the tax code, please let me know!]

      Which leads me to my confusion — when Ian writes: “It is easy to construct a tax schedule with a rising marginal rate everywhere which nonetheless redistributes in relative terms from poor to rich,” I am not following him. I am not certain if “redistributes in relative terms from poor to rich” is equivalent to “an average tax rate that, for some range of incomes, decreases with rising income”, or if, in fact, Ian means something else.

      I am missing something, but am not certain what it is! Help? Please?


      • Ian Preston Says:


        Yes, your first paragraph is exactly what I was trying to say (though mmh53b doesn’t really have a definition).

        The sort of counterexample I had in mind would be something like the following. Suppose the tax schedule has the form, say,:


        (and incomes x are bounded from below by sqrt(2) to keep average tax rates below 1 and marginal tax rates positive). Then the marginal tax rate is T'(x)=1/2-1/x^2 and everywhere rising and average tax rate is T(x)/x=1/2+1/x^2 and is everywhere falling (towards the same asymptotic value of 1/2 but from opposite sides). Since the average rate is falling, after-tax relative income gaps are widening even though you have a rising marginal rate. Does that work for you?

  4. nitpicker Says:

    I’ve missed you, Jim! Keep it up!

    • Jim Bales Says:

      Terry! Many thanks — I hope you and yours are well, and I am still hoping to swing out West and say hello in person!


  5. Jim Bales Says:

    [I think Ian and I have hit the limit on how deep WordPress will let us drive a thread. This is in reply to his example tax rate above.]

    That is the example I needed!

    I note that example incorporates an exemption (in this case, the
    first 1.414 earned is untaxed). Is that required to get the result of a falling average tax rate? [My intuition says that it is, as it lets one spread out the taxes of the marginal dollar over the untaxed portion, but my intuition has failed me before!]

    I feel that more follow-up is required from me, but must think about it before I post.


    • Ian Preston Says:

      Is that required to get the result of a falling average tax rate?

      Honest answer: I don’t know without thinking about it. My intuition – equally fallible, I’m sure – suggests we must be able to play with the schedule to get the starting point as close to zero as we want but I don’t know if we can actually drive it to zero. Obviously the tax function already described has the properties needed over the whole range x>0 but weird things start to happen if you go below 1.414 – tax liabilities exceeding income, marginal tax rates going negative and so on. We need to be clear exactly what class of functions we’re allowing ourselves to work with.

      I think the example as it is makes clear that basing a definition of progression on a rising marginal rate is not a good idea, though.

      • Jim Bales Says:


        I will, in my copious free time [/sarcasm], play around with some functions myself, and let you know what (if anything) I learn.

        And, yes, your example makes clear that defining a progressive tax as having a rising *average* rate captures the desired behavior (or “behaviour”, given your .uk domain), while defining a progressive tax as having a rising *marginal* rate casts too broad of a net, admitting tax schedules that the other definition does not.

        I deeply appreciate the counter-example, and the correction to my error.


  6. Jim Bales Says:

    Ian noted that both mmh53b and I were incorrect in our definitions of a progressive tax, and the critical measure is a rising *average* tax rate, not a rising marginal rate.

    So, I have reread my original post, asking myself if my charge — that Ms McArdle was not arguing in good faith — stands, given my error in the definition of a progressive tax. I believe that my charge stands.

    In particular, Ms McArdle responded to mmh53b with unsupported assertions from authority, stating:
    This has, contra your belief, always been a definition that characterizes a system as “progressive” rather than “regressive”.

    Notice that Ms McArdle gave no references to support her assertion that the progressiveness of a tax depends on the fraction of revenue derived from “the top” (whatever “the top” might be). And, in fact, the definitions (cited in my post) are silent as to the distribution of the revenue across the various income brackets. Clearly, Ms McArdle made no effort to check to see if the definition of a progressive tax was based on the tax schedule or the distribution of revenue across the income brackets, much less to seek out the finer distinction Ian pointed out, between marginal and average rates.

    Nor did she acknowledge (much less respond to) the counter-example mmh53b put forward, a tax where both the marginal and average tax rates decreased with income, with the effect that the vast quantity of tax was paid by the top ~0.1%. The example showed that a Ms McArdle’s definition would mis-categorize a clearly regressive tax as progressive. One arguing in good faith would have responded directly to this counter example. Ms McArdle was silent.

    In short, Ms McArdle made no effort to check that her assertions were right, she used those unsupported assertions to silence a critic, and she ignored the specific counter-example that disproved her contention. Her comment is a good lesson in how to argue in bad faith.

    Jim Bales

  7. Tom Says:

    Can I just say how much I love this comment thread. Thanks, Ian and Jim.

    • Jim Bales Says:

      Are you kidding — this is fun! (I started to explain why to my wife. She made it clear that, while she is glad that I’m having fun, hearing about it is not *her* idea of fun. Fair enough!)

  8. Jim Bales Says:


    I think I see where the weirdness comes into play (and my intuition of it coming from the existence of the zero-tax region is not correct).

    Consider the transition in tax as we move from an income of x=[sqrt(2) – epsilon] to x=[sqrt(2) + epsilon]. For any epsilon, the tax goes from 0 to 1.414. Thus, the derivative diverges, giving the marginal tax rate a delta function. (Which means, BTW, that the marginal tax rate is not monotonically decreasing, thanks to this one point.)

    To say it another way, the notion of a marginal tax breaks down at x = sqrt(2) for the original tax schedule, as the incremental income that takes one across the threshold exposes *all prior income* to taxes for the first time.

    I think the next question to ask is:
    “Is the property of decreasing average tax rate and concurrently increasing marginal tax rate a property of the discontinuity in the tax schedule or a property of the shape of the tax schedule at higher incomes?”

    Let us construct a new tax schedule that removes the discontinuity but preserves the shape of the function. In particular, let me suggest:
    T(x) = x/2 + 1/x – sqrt(2) for x > sqrt(2), and
    T(x) = 0 for x <sqrt(2).

    Now we get a tax schedule that is continuous, which insures no delta functions in the marginal tax rate.

    The marginal rate is the same (but without the delta function), i.e.,
    Marginal_Rate = 0 for x sqrt(2),

    which is monotonically increasing for all x > sqrt (2).

    The average rate becomes
    Average_Rate = 0 for x sqrt(2),

    which (if you graph it) is zero for x sqrt(2).

    We haven’t changed the shape of the tax curve, but we did get rid of the discontinuity in the tax schedule, and that was enough to get the average rate change from regressive to progressive.

    (Note that the average tax rate is a global property of T(x) while the marginal rate is a local property. So changing the marginal rate at one value of x can, in fact change the average rate for all higher values of x.)

    Your comments on the above are appreciated!

    Additionally (if, Ian, you aren’t sick of this topic already), let me revise my original conjecture:

    If the marginal tax rate is finite at zero income, and monotonically non-decreasing for all x>0, then the average tax rate must be monotonically non-decreasing.

    If anyone can think of a counterexample, I’d love to see it!


    • Jim Bales Says:

      Hmmm – a few keys lines were truncated from my comment.

      For the modified tax schedule I proposed:

      The marginal rate is the same (but without the delta function), i.e.,
      Marginal_Rate = 0 for x sqrt(2).

      This is monotonically increasing for all x > sqrt (2).

      The average rate becomes
      Average_Rate = 0 for x sqrt(2).

      This (if you graph it) is monotonically increasing for x > sqrt(2).


      • Jim Bales Says:

        OK, third try!

        One piece not getting posted is the value of the marginal tax rate for x> 1.414, which is unchanged from the case Ian put forth, i.e., 1/2 – 1/x^2

        The other is the value of the average tax rate for x>1.414, which is Ian’s case minus 1.414/x.

        I.e, 1/2 + 1/ x^2 – 1.414/x

    • Ian Preston Says:


      Sorry – not checking the internet yesterday.

      I think this paper is probably useful. Increasing marginal tax rate is obviously the same thing as convexity. Increasing average tax rate is the same thing as being `star-shaped’ (see Lemma 3). A continuous convex function which vanishes at zero must be star-shaped (see Theorem 6). (A star-shaped/progressive tax function is one where the slope of a chord to the origin is increasing. If the graph of the function passes through the origin then it is visually clear that it has to be convex, I think.) The requirement that the tax function vanish at zero can I think be relaxed to being non-positive and this is still true. A tax function with increasing marginal rate and declining average rate must therefore either have a positive tax payment at zero income or be discontinuous somewhere.

      The requirement that we don’t have positive taxes at zero income is critical, I think. The function T(x)=x+1/(1+x) has the properties of the counterexample you ask for since the marginal tax rate T'(x)=1-1/(1+x)^2 is “finite at zero income” since T'(0)=0, is “monotonically non-decreasing for all x>0” yet the average tax rate T(x)/x=1+1/x(1+x) is not “monotonically non-decreasing” – the problem is that T(0)>0.

      • Jim Bales Says:


        Many thanks for the link! I had suspected that this is a topic (relationship between marginal and average tax rates) that was examined in depth already, but I was not finding the literature.

        Am I right in claiming that the first example has the property of increasing marginal tax rate for x>1,414 yet declining average tax rate for x >1.414 because of the step change in the tax schedule?


        PS — I have no expectation of anyone checking the internet regularly, nor do I consider you to be obligated to continue this conversation once it stops being of interest — no apology needed! However, your knowledge, comments, insights, suggestions, and experience are deeply appreciated. My thanks!


      • Ian Preston Says:

        Am I right in claiming that the first example has the property of increasing marginal tax rate for x>1.414 yet declining average tax rate for x >1.414 because of the step change in the tax schedule?

        I only put the restriction x>1.414 in the first example to keep average and marginal tax rates between 0 and 1. The function itself is well-defined for x>0 without satisfying those restrictions and has increasing marginal rate and falling average rate over the range 0<x<1.414 too. Even if we want to keep the function appropriately bounded for x<1.414 we don’t have to make it discontinuous — if we define it as T(x)=x for x<1.414 then there is still a globally weakly declining average rate but the marginal rate is discontinuous at the critical point and jumps down there from 1 to 0 so is not monotonic across the whole of x>0.

        If the average rate is declining then it must be that xT'(x)<T(x) which is clearly incompatible with T'(x)>0 and T(x)=0 at any x>0. If you’re going to start at T(0)=0 and want the average rate declining across the whole range over which T(x) is positive then I think that must therefore imply an initial jump up. Is that what you have in mind?

      • Jim Bales Says:

        Ian writes:
        “A tax function with increasing marginal rate and declining average rate must therefore either have a positive tax payment at zero income or be discontinuous somewhere.”

        Yes — thank you! This is what has been bothering me.

        What got me thinking about the discontinuity in T(x) was trying to understand how (in the original example), at x=1.41, a taxpayer suddenly owed tax if the marginal tax rate was 0 for x<1.414 and 0 for x=1.41.

        It seems to me that the properties:
        1) T(0) = 0, and
        2) T(x) is continuous,
        are, in fact, characteristic of the US income tax structure. (Caveat, there are discontinuities if one looks at the tax tables, but the steps are small compared to the tax paid, and I take them to be a reasonable approximation of a continuous function.)

        While this may not be the normal course of discussion in a comment, my thanks for the impromptu lesson! Should you find yourself in Cambridge (MA), please drop by Strobe Alley at MIT and allow me to give you the tour!



  9. Kevin Donoghue Says:

    The relevant Samuelson was Paul, not Robert. (Maybe you’re directing a jibe of some sort at the Newsweakie, but it’s too subtle for me?)

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