## Greg Mankiw... Sigh: The Inestimable Value of the Samuelsonian Triad

### Greg: Ahem! DRAW THE GRAPH! Please...

Greg Mankiw: Paul Krugman... Sigh: "Paul says I have never admitted to making a math error. Well, I would if I thought I made such an error. I make them all the time. But in this case I am not convinced. Neither is University of Chicago professor Casey Mulligan, who thinks Paul made a math error. I spoke with several other economists (some of whom share Paul's politics) and they don't see Paul's point either..."

It's probably no use, given my lack of success here with:

But one more time...

Greg: You are doing a calculation. You are investigating the consequences of dropping the tax wedge τ0 between the pre-tax rate of profit (r* + τ0) and the after-tax rate of profit r* if the supply of savings is infinitely elastic at the constant after-tax rate of profit r*. You are then calculating how this drop Δτ in the tax wedge changes the amount of money going to (a) the government, (b) workers, and (c) domestic and foreign investors. You are then reporting the ratio of (a) to (b).

On a graph, this is the analysis:

In the full analysis, if we throw away the triangle area x as too small to worry about, we find:

• tax revenue changes by minus area q plus area v: -k0Δτ + τ0Δk:

• wages change by plus area q: k0Δτ

• foreigners' profits—and thus the wedge between GDP and national income—changes by +r*Δk

and the ratio of the change in government revenue to the change in wages is:

### "Static" Analysis

You, however, say you want to do a "static" analysis. A "static" analysis throws away the area v = τ0Δk, according to the old convention followed by OTA, JCT, and CBO. Why did they adopt this old convention? Because they had found that their partisan political masters always had strong views of area v, and would tell them to calculate that area v as much too large to be credible. They thought could do better in a bias-variance tradeoff sense by simply throwing away v given that the estimates of it were going to be raw political noise.

If you do that—carry out a "static" analysis—then total income does not change, and foreigners' income does not change. So _the ratio of wages gained to tax revenue lost has to be equal to:

$-\left(\frac{k_{0}Δτ}{k_{0}Δτ}\right) = -1$

There is no place else for income to go.

Income has to go to investors, workers, or the government. We have assumed that income going to investors isn't changing—that's what elastic savings supply buys you. Thus what comes out of the government must go into wages. The gain in wages cannot be any bigger—for that, you would need area v, and you would need to undertake a dynamic analysis. That should be easy to see: national income constant; after-tax returns to investors constant; what government revenue loses worker wages gain, for there is nobody else.

Yet you calculate that the "static" calculation ratio of revenue lost to wages gained is:

Why? Because you calculate area q inconsistently twice—once when you are increasing wages, and once when you are reducing revenue—and you calculate it inconsistently. Why inconsistently? It is, as Paul Krugman says, an easy-to-make algebra error. But there is a deeper root to the algebra error: you didn't draw the graph. Instead you just messed around with symbols. Alfred Marshall started us down the road of drawing lots of graphs for a reason.

Now I am not so smart. I did not draw the graph. And so it took me 4 hours on an airplane flight to figure out what had gone wrong—why you had reached a conclusion that made no sense to you: you did write:

I must confess that I am amazed at how simply [i.e., 1-t] this turns out. In particular, I do not have much intuition for why, for example, the answer does not depend on the production function...

You do not have much intuition because you cannot. Intuition leads you to one of two answers:

• for the "static" case: -1

• for the full "dynamic" case:

I do not say how any understanding of the problem as opposed to cranking-through-algebra can produce your answer , for that answer is wrong. It makes no sense to claim that area q is smaller by this factor when subtracted from taxes than when added to wages.

Draw the graph. You will see.

### The Main Point: The Samuelsonian Triad

And with that I can move on to the main, important point of this post:

This argument here has led me to change my mind on an issue of some importance.

For some time now—decades—I have been complaining that Paul Samuelson trapped us economists in a cage in which we are trying to teach our undergraduates much too much. I have been saying that Samuelson‘s curriculum was absolutely fine for the 24-year-old G.I.-Bill MIT undergraduates he designed it for in the aftermath of World War II. But I have also been saying that it is just too complex for American college students today. We ask them to:

• Draw diagrams,
• Manipulate algebra to which the diagrams correspond via René Descartes’s analytic geometry,
• Tell consistent stories about what economic actors are doing, and what their interactions add up to.

This is the toolkit established by Alfred Marshall, as supercharged by Paul Samuelson.

I have been saying that this is just too hard. Some students will get the stories, some students will get the algebra, some students will get the diagrams, and each group will get along with the particular mode. So, I have been arguing, we should focus on teaching the one piece of the Samuelsonian Triad that works the best for the most students. We should focus on getting as many of them into as much of the groove as possible. We should leave the full triad afterwards, for advanced students.

But now you—but more so John Cochrane and Casey Mulligan—have convinced me that the Samuelsonian Triad is essential. How? Through your collective insistence that a number that is really -1 is actually , which you maintain because you refuse to use all the components of the Samuelsonian Triad: you do not draw the diagram. And not drawing the diagram makes us stupid—it certainly made me stupid over the Denmark Strait.

As I find is so often the case, Paul Krugman did put it best:

Paul Krugman (2016): In Defense of Funny Diagrams: "Brad DeLong asks a question about which of the various funny diagrams economists love should be taught in Econ 101...

...There was, clearly, a time when economics had too many pictures. But now, I suspect, it doesn’t have enough.... Pictures are often the best way to convey global insights about the economy—global in the sense of thinking about all possibilities as opposed to small changes.... Some of that machinery can be very useful as tools for clarification.... I have the sense that too many majors and/or grad students were shortchanged on this front. They can do game theory, they can solve sets of equations, but their sense of how the pieces fit together is lacking, and—at least in some seminars I’ve sat in on—too many don’t have the technique to cut through what should be easily avoidable confusion....

Draw, baby, draw...

And I now agree: the full Marshallian toolkit—the full Samuelsonian triad—is essential. For the most disturbing thing is that this is an Econ 1-level tax incidence calculation we are doing here. And, without going through the repetitive formal steps of the Samuelsonian triad, tenured economics professors at major research universities cannot reach consensus agreement on what the right answer to Econ 1-level calculations are. That means that we are in huge trouble.

So never mind that students find it difficult to do the argument once in words, once in algebra, and once in graphs. Never mind that students—even pretty good students—will glom on to one of the three, and fail to understand how they relate. Never mind that you have to be very good at all of words, algebra, and graphs in order to get the interrelationships right. Never mind the number of exams I read in which the algebra is right, but the graph is wrong. Never mind the number of exams I read in which the words are right, but the algebra is wrong. We have to require that everybody starting out attempt, at least, to master all three parts of the Samuelsonian triad. Why? Because all three parts are necessary to serve as checks on one another, and to try to deal with the fact that we are all, individually, Bears of Very Little Brain.

### Keynes's View

All this throws me back to John Maynard Keynes's obituary for his professor Alfred Marshall:

Alfred Marshall... [was] endowed with a double nature.... As a preacher and pastor of men he was not particularly superior to other similar natures. As a scientist he was, within his own field, the greatest in the world for a hundred years....

In [this second] respect the diversity of his nature was pure advantage. The study of economics does not seem to require any specialised gifts of an unusually high order. Is it not, intellectually regarded, a very easy subject compared with the higher branches of philosophy and pure science? Yet good, or even competent, economists are the rarest of birds.

An easy subject, at which very few excel!

The paradox finds its explanation, perhaps, in that the master-economist must possess a rare combination of gifts. He must reach a high standard in several different directions and must combine talents not often found together. He must be mathematician, historian, statesman, philosopher-in some degree. He must understand symbols and speak in words. He must contemplate the particular in terms of the general, and touch abstract and concrete in the same flight of thought. He must study the present in the light of the past for the purposes of the future.

No part of man's nature or his institutions must lie entirely outside his regard. He must be purposeful and disinterested in a simultaneous mood; as aloof and incorruptible as an artist, yet sometimes as near the earth as a politician. Much, but not all, of this ideal many-sidedness Marshall possessed. But chiefly his mixed training and divided nature furnished him with the most essential and fundamental of the economist's necessary gifts-he was conspicuously historian and mathematician, a dealer in the particular and the general, the temporal and the eternal, at the same time...

How is this different from, say, physics?

In physics, the math that fits reproducible experiments very well rules. Get the math—or a math—down so that it fits experimental results, and then there is no appeal from it. The task then is to understand what the math is telling us. The most famous historical example of this are the Maxwell-Gibbs-Heaviside-Hertz equations, which Maxwell wrote down in 1861 and which the other three rewrote as four vector equations in 1884. Inside of them, Einstein's theory of special relativity is yelling at physicists: "HERE I AM!" Yet it was not until 1905 that Albert put his finger on what the equations were really saying:

Maxwell's electrodynamics... applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor.... If the magnet is in motion and the conductor at rest, there arises... an electric field with a certain definite energy, producing a current.... But if the magnet is stationary and the conductor in motion, no electric field arises.... In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise... to electric currents of the same path and intensity.... Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “luminiferous ether”, suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest...

But this cannot work in economics: the models and the math do not fit reproducible experiment precisely, but are instead only weak crutches on which to try to walk forward. We cannot just write down equations that fit and try to tease out what they are telling us. We have to ask whether they are the right equations to write down...

And now we come to the part that makes me sad. Mankiw writes:

I am not convinced. Neither is University of Chicago professor Casey Mulligan, who thinks Paul made a math error...

Greg: Casey Mulligan is not a very good economist. You know that he is not a very good economist. You have read his writings as much as I have. For example:

The only rational conclusion is: ideologically-blinded unprofessional incompetence. Don't cite Casey Mulligan as an authority on anything. (And it is also unwise to cite the lurkers who support you in email: that trick never works.)

So let's look at Casey Mulligan's claim that it is "Paul [Krugman who] made a math error". Does it fare any better than his October 2008 claim that only those who worked in finance needed to be afraid?:

This morning I tried to help Mr. Krugman with this by posting on his twitter:

What @delong is seeing is that "static" revenue loss is arbitrary:

the static revenue loss from a per-unit tax cut [what Krugman shows with his blue rectangle and calls "direct"]

is different from [Furman's] static loss from an ad valorem tax cut [what a corporate-income tax cut would be],

even when those cuts are scaled so that both have the same effects on revenue and the surplus of all parties.

That is why I use actual revenue loss.

The ratio between the two "static" concepts is the (1-t) factor that has Delong and Krugman so confused.

No, Casey, "static" is not "arbitrary". It is not whatever you define it to mean.

As I wrote before: "Static" is a term of art used by the revenue estimators at Treasury's OTA and Congress's JCT (and those who shadow and check and dispute their calculations at CBO, OMB, TPC, and a host of other alphabet-soup Washington organizations).

When Greg uses the word "static", he is referring to them and their procedures. He is claiming that he is—in his model—doing what they do.

When OTA and JTC do revenue estimates, they do not take the change in the tax rate and multiply it by a frozen number that is the tax base. They allow for asset prices and returns to shift in response to changes in tax rates. Those shifts change the tax bases on which the new tax rates are levied. They allow for people to shift from one activity to another and to relabel their income streams, thus changing the tax bases yet again. What makes the estimate static is that the Jedi Masters of Revenue Estimation do not let themselves be bullied by their overoptimistic political masters into making estimates that are wildly overaggressive in their assessments of investment and growth.

If there were no OTA or JCT—and if the stakes at risk did not include the credibility and influence on policy of the reality-based technocrats at OTA, JCT, & co., I would be very tempted to say to the circus monkeys: Go have your circus! Define "static" however you want! Know that your calculations do not correspond to any areas on any sensible economics graph! Know that they do not correspond to anything in any coherent assessment of incidence! Know that what you are doing is the equivalent of looking at your genitals over and over again! But it is a free country—do what you want!

The problem is that there are serious people who do this tax incidence stuff for a living at OTA, and JCT, and elsewhere. We all owe them a great deal.

I am, perhaps, more knowledgeable about what they do, and respect them more, as a result of my time spent working downstairs from them in the Treasury.

I am, perhaps, more honorable, in that I think that the work of competent people trying to do their best to make America greater should not be casually trashed via misrepresentation by ignorant circus monkeys.

But maybe that's just me...

And then there is another sad part: John Cochrane was once a good economist. Now:

Each dollar (per worker) of static tax losses raises wages.... For t=1/3, each dollar of tax cut raises wages by 1.50 dollars. A number greater than one does not mean you're a moron, incapable of addition, a stooge of the corporate class, etc. The example is gorgeous, because all the production function parameters drop out.... The question is not whether one dollar of static tax cut produces more than a dollar of revenue. The question is whether it raises capital enough to produce more than a dollar of wages. This is also a lovely little example for people who decry math in economics. At a verbal level, who knows? It seems plausible that a 1 dollars tax cut could never raise wages by more than 1 dollars. Your head swims. A few lines of algebra later, and the argument is clear. You could never do this verbally...

No, John. "You could never do this verbally" not because math is powerful, but because you are wrong. In a static analysis national income says constant. What comes out of taxes has to go into wages, because the model assumes it doesn't go into profits, and the government's taxes, workers' wages, and investors' profits are all that there is. The coefficient is -1, by the metaphysical necessity of the case.

DRAW THE GRAPH!

Now that Cochrane and Mulligan are mired in numerical error does not disturb me. I have seen them in action to know that they stopped thinking—stopped being anything I would call a professional economist—long ago. (In Cochrane's case, consider his claim last week: "So, what's up with Bitcoin? Is it a 'bubble?' A mania of irrational crowds?... Bitcoin seems to me like a perfectly 'normal' phenomenon. Intersect a convenience yield and speculative demand with a temporarily limited supply..." when there has been no noticeable increase in the prices of any of the other assets with convenience yields analogous to that of BitCoin.)

Mankiw is much more disturbing: he really should by now have rethought the issue, tried hard to gain some intuition about where his factor , drawn the graph... and discovered that he got it wrong.

That he has not is what has driven me away from my old position. It has driven me to Paul Krugman's "Draw, Baby, Draw!" The Marshallian toolkit—the Samuelsonian triad—is, I now think, an intellectual inheritance of inestimable magnitude. It is hard to master. But we have to do so, lest our failure to do so make us stupid.