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IS-LMic Extremism (With a Postscript on the Nobel Prize)

Published
by
Steve Landsburg
on October 10, 2011
in Economics
. 37 Comments

The IS-LM model is the simplest version of the “old Keynesian” approach to macroeconomics. You don’t hear much about IS-LM in the research literature these days, but it’s been coming up a lot on the blogs. Tyler Cowen tells us what he doesn’t like about the model; Brad DeLong and Paul Krugman rise to its defense; Stephen Williamson takes up the gauntlet, and Scott Sumner weighs in.

None of them, in my opinion, has touched the main issue, which is that IS-LM provides absolutely no framework for policy analysis because it makes no assumptions, and draws no conclusions, about what people are trying to accomplish. If you don’t know what people are trying to do, you can’t possibly know how best to help them.

Suppose, for example, that, as Paul Krugman believes, the current state of the economy is being driven by a “liquidity trap”, which means that people are hoarding money instead of spending it, and therefore consuming less, which is why employment is so low. Two weeks ago on this blog, I posed the following question to the IS-LMers:

Why aren’t you thrilled with the current state of the economy? … Why, as the stock of money continues to grow, shouldn’t the joy of hoarding eventually compensate for the annoyance of not having food on the table?

The IS-LM model provides no answer to that question. A model that can’t decide whether the current US economy is in a state of Nirvana is not a useful model for policy evaluation.

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Double Star!

Published
by
Steve Landsburg
on October 7, 2011
in Silk
. 3 Comments

This one makes me happy!

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Stopped Clocks

Published
by
Steve Landsburg
on October 6, 2011
in Blogging, Current Events, Paul Krugman and Politics
. 33 Comments

Incidentally, Paul Krugman made an incisive point last week when he wrote:

Here’s a question I haven’t seen asked: If fear of future regulations and taxes is holding business back, as everyone on the right asserts, why didn’t the Republican victory in the midterms set off a surge in employment?

After all, if you really believed that fears of Obamanite socialism were the key factor depressing employment, the GOP victory — with the clear possibility that the party will take the Senate and maybe the White House next year — should greatly reduce those fears. So, where’s the hiring surge?

I even set out to write a blogpost citing this argument with approval — but around the time I was composing it, Krugman followed up with this bit of idiocy, to which a response seemed more urgent.

Now that that’s out of the way, I can come back to the bit about the missing Boehner Boom. It’s a more-than-fair question. How would you respond to it?

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Half Scorpion

Published
by
Steve Landsburg
on October 5, 2011
in Silk
. 10 Comments

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Big News

Published
by
Steve Landsburg
on October 4, 2011
in Logic, Math, Progress and Truthseeking
. 51 Comments

Last week, the highly distinguished Princeton Professor Ed Nelson announced a proof that the Peano axioms for arithmetic are inconsistent — and hence so is arithmetic itself. If true, this would be much bigger news than faster-than-light neutrinos. It would be bigger news than a discovery that the South had won the American Civil War. It would be far, far bigger news than a discovery that all life on Earth was intelligently designed.

There are, after all, multiple proofs that Peano Arithmetic (that is, the fragment of arithmetic described by the Peano axioms) is consistent. Among those, the simplest and most convincing (to the overwhelming majority of mathematicians) is this: The axioms of Peano Arithmetic, and therefore the theorems of Peano Arithmetic, are all true statements about the natural numbers — and a set of true statements cannot contradict itself.

Ed Nelson rejects that argument because (exempting himself from that overwhelming majority) he doesn’t believe in the set of natural numbers — or perhaps even in individual numbers when those numbers are very large. (How do you know that 810000 exists? Have you ever counted to it?)

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There He Goes Again

Published
by
Steve Landsburg
on October 3, 2011
in Bad Reasoning, Economics, Paul Krugman, Policy and Politics
. 84 Comments

Paul Krugman’s latest venture into self-parody starts with a recent paper on the cost of air pollution, which finds that said costs are big and heavily concentrated in a few industries. Krugman then links to a New York Times article surveying Rick Perry’s past clashes with the EPA. With no further argument, he concludes that

Today’s American right doesn’t believe in externalities, or correcting market failures; it believes that there are no market failures, that capitalism unregulated is always right. Faced with evidence that market prices are in fact wrong, they simply attack the science.

Where to begin?

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Happy Birthday

Published
by
Steve Landsburg
on September 30, 2011
in Anniversaries, Birthdays, Math and Progress
. 1 Comment

twoMathOverflow turns two years old this week — a milestone in the transformation of mathematical research into a massively collaborative endeavor. It’s happening on blogs, it’s happening on mailing lists, and it’s happening in a big way on MathOverlow, where mathematicians ask and answer the sorts of questions that might come up in the faculty lounge — if the faculty lounge were populated by hundreds of experts pooling their expertise.

If you’re interested in mathematics at the research level, MathOverflow is a place to learn something new and fascinating every single day. (If you are not doing mathematics at a research level, feel free to read but please don’t feel free to join the fray; questions at anything below about a second-year graduate level should be directed to MathStackExchange, another massively collaborative site aimed, roughly, at the college level — which reminds me that it’s not just mathematical research, but also mathematical education, that is being revolutionized before our eyes.)

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D’oh!

Published
by
Steve Landsburg
on September 28, 2011
in Economics
. 39 Comments

Well, it took embarrassingly long for me to see this but there’s really a very simple resolution to the quandary I posted Monday. The key point is this:

In a flexible price world, anybody can supply money at a social cost of zero. In a fixed price world, only the government can.

To be more precise: I currently hold $11. Suppose I agree to hold a twelfth. In a flexible-price world, I can get this dollar from the government (which prints it up at zero cost) or I can get it from my friend Jeeter, whereupon the price level adjusts and the rest of the world’s real balances (including Jeter’s) are restored to their original level. No social cost either way. In a fixed-price world, I can get this dollar from the government (which prints it up at zero cost) or I can get it from my friend Jeeter, whereupon prices don’t move and the rest of the world’s real balances are reduced. Zero social cost one way, positive social cost the other way.

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A Keynesian Quandary

Published
by
Steve Landsburg
on September 27, 2011
in Economics
. 64 Comments

Last week I blogged about my perplexity regarding the Keynesian notion of a liquidity trap.

In thinking about this harder, I’ve come to realize that a good part of my confusion has nothing to do with liquidity traps. It comes down to a very specific question about sticky-price models in general.

I expect this discussion will be interesting only to the most wonkish of my readers. The non-wonkish are invited to ask for clarifications, but please don’t jump in claiming to have “definitive” answers unless you’ve got a good grasp of the basics. I’d like to keep this discussion on track, and I’d like to learn something from it. Uninformed noise will be counterproductive.

For what it’s worth, I’ve discussed this offline, at considerable length, with several very good macroeconomists who eventually pronounced themselves as confused as I am. I really am hoping somebody with the right insight will pop up here and set us all straight.

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Neutrinos and Appomattox

Published
by
Steve Landsburg
on September 26, 2011
in Current Events, Musings, Physics, Science and Truthseeking
. 56 Comments

Scientists at CERN have found apparent evidence that neutrinos can travel faster than light.

Suppose that tomorrow historians at Harvard find apparent evidence that the South won the American Civil War — not in some metaphorical “they accomplished their goals” sense, but in the literal sense that it was actually Grant who handed his sword to Lee at Appomatox and not the other way around.

Question: Of which conclusion would you be more skeptical?

Of course your answer might depend on exactly what this new “apparent evidence” consists of. So let me reword: As of this moment, which do you think is more likely — that neutrinos can travel faster than light, or that the South won the Civil War?

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Are These The Good Old Days?

Published
by
Steve Landsburg
on September 23, 2011
in Economics
. 102 Comments

I am not well versed in Keynesian business cycle theory. Therefore I have a very naive question for the Keynesian economists:

Why aren’t you thrilled with the current state of the economy?

Here’s why I ask: According to what I take to be an orthodox Keynesian view, we are now in a liquidity trap. (My question does not apply to Keynesians, new or old, who believe otherwise.) That means that people want to hold lots and lots of money instead of spending it. Cool! We can provide money at almost zero cost. So it should be easy to make people very happy. What’s the problem?

Of course, people are working less, but that makes perfectly good sense in a world where people prefer to consume less. Why spend all day on an assembly line churning out widgets that people prefer not to buy?

A quick and obvious answer is that the people who are choosing to accumulate money and the people who are out of work are not the same people. In other words, to put this in slightly more technical language, you can’t address this question in a so-called “representative agent model” — a model that abstracts from interpersonal differences.

Still: The theory, as I understand it, is that vast numbers of people are choosing to hold vast amounts of money. Since money can be produced costlessly, this ought to count as a very good thing — which should offset a lot of very bad things, no?

Whatever answer there is might vary from one Keynesian economist to another, so let me subdivide my question into two:

  1. Why aren’t “old Keynesians” perfectly happy with the current state of the economy?
  2. Why aren’t “new Keynesians” perfectly happy with the current state of the economy?

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A Tale Told By an Idiot

Published
by
Steve Landsburg
on September 22, 2011
in Books, Favorites and Memories
. 21 Comments

In sixth grade, I did not read My Side of the Mountain, though it was assigned for class. In eighth grade, I did not read Little Women and in ninth grade I did not read Great Expectations and The Good Earth. As I passed through high school, I worked my way through much of the western canon, not reading The Scarlet Letter, Bartleby the Scrivener, The Return of the Native, and dozens more. In eleventh grade, we were assigned two books by Steinbeck, two by Hemingway, two by Sinclair Lewis and two by William Faulkner. I did not read the Steinbeck, Hemingway or Lewis but for some long-forgotten reason I violated years of established tradition by tackling the Faulkner — specifically As I Lay Dying and The Sound and the Fury.

As I Lay Dying went down pretty easily, but I remember many nights struggling my way through The Sound and the Fury, Cliff notes at my side. It felt like scaling Everest, and the vistas at the top were worth the climb.

A couple of weeks ago, as part of my ongoing project to read great novels, I decided to revisit The Sound and the Fury, and I’m more than glad I did; I finally have an answer to give the next time I’m asked what one novel I’d bring to a desert island. But what I’m flabbergasted by is this: How did this book ever get assigned to high school students in the first place? I ask for at least two reasons:

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Survival of the Fittest

Published
by
Steve Landsburg
on September 21, 2011
in Current Events and Economics
. 37 Comments

The Wall Street Journal reports that `daily deal’ sites like Groupon are dying fast, casualties of the expensive competition for new users. Groupon now spends about $8 to lure one active user.

This looks like a good example of Darwinian competition yielding an inefficient outcome — as we should expect. (See Chapter 8 of my book The Armchair Economist for more on why Darwinian competition is nothing like market competition, and far more likely to yield bad outcomes.) Vast sums are being spent in an arms race with relatively little social value. Surely consumers benefit from all this competition, but it’s highly implausible that they benefit enough to justify such high expenditures. Even after the recent Great Winnowing, about 350 of these sites remain; surely consumers (none of whom have the time to visit 350 sites a day) would be almost equally well served by 50 sites, at about 1/7 the cost.

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Compassion Play

Published
by
Steve Landsburg
on September 19, 2011
in Current Events, Economics, Morality, Outrage, Paul Krugman and Policy
. 83 Comments

One thing I like about the study of economics is that it fosters compassion. When part of your job is to predict human behavior, you quickly learn the value of understanding other people’s problems. When the other part of your job is ferreting out the unseen global consequences of our choices, you’ve taken the first step toward caring about those consequences.

For example: Suppose a guy with no health insurance and no assets shows up at a hospital emergency room with an urgent life-threatening condition. Should you let him die? Ordinary compassion says no. The heightened compassion of the economist says, at the very least, maybe.

First, a policy of providing emergency health care to everyone is pretty much the same thing as a policy of providing emergency health insurance to everyone. It was specified here that this was a guy who didn’t want health insurance. So let’s recognize for starters that such a policy runs counter to — I am tempted to say runs roughshod over — the guy’s own revealed preference. It’s an odd sort of compassion that forces people to buy things they don’t want.

Now you might object that nobody’s forcing this guy to buy emergency health care; we’re trying to give him emergency health care. Not so fast. Here’s the first place where a little economic training goes to hone one’s sense of compassion: The emergency health insurance we’re foisting on this guy has a cost. We can spend that money on emergency rooms or we can spend it on a myriad of other things the guy might prefer. How is it compassionate to give him one thing when he prefers another?

This is particularly true if the guy happens to be very poor. Poor people have a lot of problems, and emergency health care is only one of them. They need better education, they need better transportation, and they need a little help buying groceries.

There is room for lots of debate and lots of disagreement about how much we as a society should be spending to help poor people. That’s not the issue here. The issue here is: Given that you’ve decided to spend an extra such-and-such many dollars a year helping poor people, why would you spend it in this particular way rather than one of the many other ways they could use it? For God’s sake, why not at least ask them if they’d rather have the cash?

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Cats, Dogs and Quantum Mechanics

Published
by
Steve Landsburg
on September 13, 2011
in Economics, Game Theory, Math and Physics
. 83 Comments

The game of Cats and Dogs works like this: You and your teammate are placed in separate rooms and forbidden to communicate. You are each asked a randomly chosen question: Either “Do you like cats?” or “Do you like dogs?” (Each of your questions is determined by a separate fair coin flip.)

You win if your answers agree — unless you were both asked the “cats” question, in which case you win if your answers disagree.

A little reflection should convince you that if you are allowed to meet with your partner and plot strategy before the game, then the best you can do is agree to always agree — say by both always answering “yes”. That way, you win 75% of the time, and there’s no way to do better. In particular, there’s nothing to be gained by randomizing your answers.

That, at least, is true, in a world governed by the laws of classical physics and probability theory. But in a world governed by the laws of quantum mechanics — which is to say, in the world we live in — you can in principle do better. Namely: You each carry with you one of a pair of entangled “quantum coins” (actually elementary particles, but I prefer to think of them as coins, since you’re going to use them as randomizing devices).

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Thursday Puzzle and More

Published
by
Steve Landsburg
on September 8, 2011
in Current Events, Economics, Politics and Puzzles
. 10 Comments

Yesterday’s post on taxation generated a whole lot of comments that deserve responses; unfortunately I’m too swamped right now to respond. Worse yet, I’ll be out of town — and probably not blogging — for the next few days. Sometime next week, I’ll try to craft a new blogpost addressing much of what was said in those comments.

Meanwhile, here, courtesy of our frequent and invariably interesting commenter Mike H, is a puzzle to keep you busy while I’m gone:

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The Romney Plan

Published
by
Steve Landsburg
on September 7, 2011
in Current Events, Economics and Politics
. 44 Comments

I have not read or even skimmed Mitt Romney’s 160-page economic plan; all I know is what I’ve seen in the headlines. So all of this is subject to revision. But:

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Moral Matters

Published
by
Steve Landsburg
on September 5, 2011
in Morality and Philosophy
. 67 Comments

The ever-insightful philosopher Peter Smith has a number of interesting things to say about abortion, but I found one of those things particularly striking — partly because I don’t recall ever having thought of it before, and partly because, in retrospect, I don’t see how I could have failed to think of it.

Namely: The argument is made that zygotes/embryoes/fetuses, even at a very early stage, have the full moral status of human beings. Yet if that were true, surely we’d want to divert a substantial portion of the medical research budget away from relatively minor scourges like, say, cancer, to the spontaneous abortions that take the lives of something like 30% of these full-fledged humans. In a typical year, there are about 8 million cancer deaths worldwide; the number of early-stage spontaneous abortions must be at least twice that.

In Smith’s words:

very few of us are worried by the fact that a very high proportion of conceptions quite spontaneously abort. We don’t campaign for medical research to reduce that rate (nor do opponents of abortion campaign for all women to take drugs to suppress natural early abortion). Compare: we do think it is a matter for moral concern that there are high levels of infant mortality in some countries, and campaign and give money to help reduce that rate.

Smith is struck by the fact that this attitude is very widespread; I am more struck by the fact that it seems to be very widespread even among those who characterize themselves as pro-life.

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Recap

Published
by
Steve Landsburg
on September 2, 2011
in Bad Reasoning, Economics, Paul Krugman and Policy
. 31 Comments

Some commenters still seem confused about the locus of disagreement in this week’s back-and-forth with Paul Krugman. I post today not to beat a dead horse, but to clarify the issues for those who are interested in understanding them. Please keep any discussion both civil and on-topic. I’ve numbered the points below for easy reference.

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Krugman Followup

Published
by
Steve Landsburg
on September 1, 2011
in Bad Reasoning, Current Events, Paul Krugman and Politics
. 31 Comments

What I like about people in academics is that when we disagree, we actually care about figuring out who’s right — and therefore we have a tendency to reach consensus, though it can take a while.

Anybody who blogs often enough (very much not excluding yours truly) is occasionally going to post something that, at least as written if not as intended, is objectively plain flat out wrong. Paul Krugman did that a couple of days ago, I responded, he’s responded to my response, and at least 4/5 of our disagreement is now resolved. That’s exactly as it should be.

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Thursday Puzzle

Published
by
Steve Landsburg
on September 1, 2011
in Puzzles
. 51 Comments

quartersOur frequent and reliably insightful commenter Jonathan Kariv sends along this neat puzzle:

Your enemy chooses 10 points on an infinite tabletop and gives you 10 coins of the same size (let’s say U.S. quarters). Can you always place the coins on the tabletop in such a way that all 10 points are covered, but no two coins overlap?

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Panglossian Economics

Published
by
Steve Landsburg
on August 31, 2011
in Bad Reasoning, Current Events, Economics and Paul Krugman
. 102 Comments

PanglossIn a radical departure from his previous expressions of dissillusionment, Paul Krugman has implicitly declared in his latest blog post that we are now living under the best of all policy regimes. I presume he will now be able to retire with satisfaction from his career as a gadfly.

The context is Eric Cantor’s demand that any federal disaster relief in the wake of Irene be offset by spending cuts elsewhere. Krugman thinks this is silly, and proves his point with an appeal to the standard Ricardian theory of public finance. According to that theory, which all economists understand and accept, if you’ve got to bear a cost, it’s best to spread that cost out over as many activities as possible. So ideally, you’d pay for disaster relief partly through spending cuts, partly through (current) tax increases, and partly through an increase in the deficit. Therefore says Krugman, “the bottom line is that basic, regular economics says that Cantor isn’t making sense.”

Since Krugman has carelessly neglected to spell out an important detail of his argument, let me fill in the gap for him: The Ricardian conclusion does not come from thin air; instead it follows logically from certain premises, key among which is that you’re starting from an ideal policy regime.

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How I Spent My Summer Vacation

Published
by
Steve Landsburg
on August 30, 2011
in Books and Favorites
. 33 Comments

ibookshelfA few years ago, I discovered that reading on my Kindle is about 1000 times better than reading a book. This year, I discovered that reading on my iPhone is about 100 times better than reading on my Kindle. As a result (and also as a result of a lot of time spent on airplanes), I’ve been on a mad fiction-reading spree the past few months. Some mini-reviews:

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Terrifying Prospects

Published
by
Steve Landsburg
on August 28, 2011
in Paul Krugman
. 48 Comments

Paul Krugman wisely reminds us that:

The odds are that one of these years the world’s greatest nation will find itself ruled by a party that is aggressively anti-science, indeed anti-knowledge. And, in a time of severe challenges — environmental, economic, and more — that’s a terrifying prospect.

Yes, a terrifying prospect — and an excellent reason to limit the powers of ruling parties, though Paul never seems to notice this.

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Travel Report

Published
by
Steve Landsburg
on August 28, 2011
in Blogging
. 1 Comment

Thanks to Hurricane Irene, my trip to Brazil turned into a trip to Dulles Airport, from which I am now (after much scrambling) returned. I hope to make it to Brazil in the near future!

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Public Service Announcement

Published
by
Steve Landsburg
on August 26, 2011
in Blogging
. 1 Comment

For the next several days, I’ll be in at the Liberty and Democracy Forum in Brazil, and likely out of Internet range. I’ll
see you late next week.

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A Consequential Study

Published
by
Steve Landsburg
on August 24, 2011
in Bad Reasoning and Morality
. 22 Comments

By now you’ve probably encountered one of the various “do you push the fat man in front of the subway train to stop it from running over five innocent people trapped on the tracks?” puzzles that moral philosophers have been using lately to distinguish the consequentialists (who care only about ends) from the deontologists (who care also about means). (I invoked some of these puzzles, in a slightly non-conventional way, in Chapters 16 and 17 of The Big Questions.)

Now come psychologists Daniel Bartels and David Pizarro to report that choosing what they call the “utilitarian” solution (by which they mean the consequentialist solution, though one might have more faith in their scholarship if they’d used the accurate word) is correlated with higher scores on measures of psychopathy, machiavellianism, and “life meaninglessness”. Those who would push the fat man are more likely to agree with statements like “I like to see fistfights” or “The best way to handle people is to tell them what they want to hear” or “When you think about it, life is not worth the effort of getting up in the morning”. In other words, they are what we tend to think of as an unpleasant bunch.

But you see, here’s the thing: Bartels and Pizarro did the wrong test. Because in every one of the dozen or so moral dilemmas presented to the subjects, they asked: Would you shoot the crewmember? or would you execute one of the hostages? or would you flip off the switch to the ventilator? — which doesn’t get to the moral issues at all. The moral issue is this: In your opinion, should you shoot the crewmember or execute the hostge or flip the switch? The researchers never asked those questions, so we have no idea what the subjects’s opinions were.

Tell me a story about 20 children trapped in a burning building and ask me if I think I should rush in to save them. My answer is yes. Now ask me if I would. If I’m honest, I am very unsure I would do the right thing.

So what do we learn from this research? Here’s one interpretation: Maybe most people agree that you should push the fat man, and most people realize that they themselves would shrink from this moral duty out of squeamishness (just as most people agree that you should save the children from the fire, and most realize they might shirk this moral duty). Therefore they answer “No, I would not push the fat man.” But it’s only the psychopaths and Machiavellians who exaggerate their own moral virtues by claiming that they would rise to this particular occasion.

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Mea Culpa

Published
by
Steve Landsburg
on August 23, 2011
in Blogging
. 30 Comments

For the first time ever, I am deleting a post.

The numbers in this morning’s post (now missing) were completely wrong as were, therefore, the conclusions I drew from them.

For the record, all of the numbers concerned what happens if you save $1000 a month. I often show these numbers to my students, and when I do, I get the assumption right. But this time, I somehow a) convinced myself that the assumption was $100 a month, not $1000 a month and b) therefore concluded that saving is a whole lot easier than what I tell my students every year, and c) said a lot of nonsense that followed from this.

(I tell my students that for *them*, saving $1000 a month will soon probably be a plausible strategy, which is likely to be true. Having conflated $1000 with $100, I drew implausible conclusions in the blog post about what you could do on $25,000 a year.)

I could say things about the folly of posting at 2AM, but I think the wiser course is simply to apologize.

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Economics 102

Published
by
Steve Landsburg
on August 22, 2011
in Economics and Paul Krugman
. 13 Comments

One of Paul Krugman’s favorite tactics is to assert that all he’s doing is channeling the time-honored lessons of Economics 101 — pre-empting dissent with the implication that any dissenter must be either an ignoramus or a radical. (Journalistic honesty compels me to acknowledge that I might have employed this rhetorical tactic once or twice myself over the years.)

It’s interesting, then, to take note of how very far his central arguments actually deviate from Economics 101. Here’s what he said last week on his blog:

Mulligan and others keep emphasizing examples of individual groups that have managed to gain jobs by cutting wages or offering other attractions to would-be employers. They then assert that these examples tell us what would be needed to expand overall employment.

The point, of course, is that all such arguments amount to committing the fallacy of composition…The essence of macroeconomics is understanding why such things are a fallacy, why what happens if one group does something is not at all what happens when everyone does it.

But you see, here’s the thing: According to the standard Economics 101 version of the sticky-wage Keynesian model, this is a case where what happens if one group does something is exactly the same as what happens when everyone does it. According to that model, as long as wages continue to fall, firms will continue to move along their labor demand curves until we reach full employment.

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Friday Solution

Published
by
Steve Landsburg
on August 19, 2011
in Math and Puzzles
. 12 Comments

Re yesterday’s puzzle, you’ll find answers in the comments. (We are blessed with some very smart commenters here at The Big Questions!!)

Commenter Roger Schlafly pointed this Wikipedia article where I was surprised and delighted to see a reference to a paper co-written by my old friend Dave Rusin. I did not remember that Dave had anything to do with this problem, but in retrospect I bet I knew this at one time.

I managed to dig out some notes I jotted down on this subject many many years ago. I have not doublechecked these results, and I can’t completely vouch for the careful accuracy of my younger self, so take these for what they’re worth. But here’s what I once claimed to have proved:

The reason there is exactly one pair of nonstandard six-sided dice is that six is the product of two distinct primes. For the same reason, there is exactly one pair of nonstandard n-sided dice when n is 10, or 15, or 21, or …. For any product of three distinct primes, there are at most 40 nonstandard pairs.

I also found (in what appears to be my handwriting) this chart, which I reproduce with the same caveats:

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