The Age of Reason

I don’t know which is scarier the height of the curve around age 50 or the slope of the curve (fyi, my guess is cohort effects are small). The slide is from David Laibson who has much more on aging and dementia; also raises issues of the value of medical care that maintains the body but not the mind.


Makes you wonder about all these people who want to raise the retirement age to balance the budget...

Not really. The slope of the curve is quite modest until the mid-late 70s. Also, I'd like to see the results for people younger than 50, which could help to untangle the efects of aging from general innumeracy.

Btw, this is yet another example of a graph where there is no reason the vertical axis should not start at zero. Major pet peeve.

Also, there is plenty reason to believe that gainful employment (paid or otherwise) can delay the onset of dementia.

Wow! I had no idea how much math education has improved since the 1940's.


There is no such thing as "The Retirement Age". There is just an age at which the government will start paying you because you're alive.

Actually, there is such a thing as the retirement age. 70 1/2. Mandatory [taxable] withdrawals.

This only applies if you have a traditional IRA, so it will be less of a constraint as people with Roth IRAs age (unless Congress decides that you can't keep your capital gains tax free after all), and there's no legal reason you can't continue to work while taking your required withdrawals. Weekend getaway every weekend?

I think the question of retirement is a real issue. And it's linked to Tyler's remark about the lack of healthcare for the mind. If we could improve the mental agility of the mind the issue could be ameliorated. I do not want to be a living cognitive vegetable. Perhaps we should legalize euthanasia?

Otherwise I don't see any good answers. Clearly older workers are going to be less efficient. And as life expectancy increases they are going to be an ever increasing burden on the rest of society if they cannot work.

I have a few suggestions to allow older people to work and thus for society to remain solvent.

1. Start college level math, science and humanities in high school (South Korea, Japan and China style.) And yes we need to revamp middle school fast.
2. Finish high school at age 17 (for you econ people, calc I, II, III, linear algebra and diff eq, intro macro & micro finished and perhaps intermediate also finished.)
2. Have a standard 4 year college degree where during the last 3 years one just pursues one's major. Get rid of post docs (they are only there because PhDs could not get jobs). Make PhDs shorter. 4 years max.
3. As a consequence get people into the labor force earlier. Lessen time for med school (and again from personal experience - a lot can be cut.) Cut the barriers to entry for most fields (like get rid of masters for accounting that was only introduced in the 90s to artificially increase wages.)
4. Explicitly legalize physical, cognitive and other testing by employers and allow age discrimination in employment.
5. Lead by example - instead of minimum ages to be president and in Congress, establish maximum ages. Eg - you cannot be president or in congress over age 50. We need a lot more 25 year olds in congress (remember McCain didn't know how to go online yet he was passing bills concerning the internet - madness.)
6. Encourage industry to always hire CEOs under age 50. Same idea for supreme court justices (why the most complicated issues in our country go before some conked out geriatrics is beyond me.) Cabinet secretaries should be hired in the 30s and early 40s.
7. Encourage low skill and menial jobs to be given to the elderly - over age 60 and especially age 70. Ie - we need to not waste having a young person being a janitor. That way we can efficiently allocate labor and keep people in the 60s and 70s working. Same idea in the professions. Start working at 22 (have some legal education in undergrad and a year or two after). Make partner at age 25. Senior partner at age 30. Peak at age 35. At 45 made junior partner. Att age 50 become associate. At age 60 become paralegal. At age 65 legal secretary (or something similar). At age 70 office assistant, janitor or whatever needs doing.
8. Mandate that all instructors in higher ed over age 50 must have cognitive testing and if they don't compare favorably then they are let go or they can become adjunct faculty. Same career path for them as for lawyers above.
9. PIs in higher Ed - max age of 55. [And I say this from personal experience! Some old fuddy duddy is useless is useless. It's people in their 20s and 30s who come up with the innovation. And it's so sad that the old fogies gobble up most of the cash for research.]
10. Establish a norm that people aged 30 - 45 are the bosses of everyone else. That way everyone realizes that once you get older, you go down the social economic scale and no-one is surprised. Careers become an asymmetric bell curve.

And indeed we should pay people in their mid 30s the most - they have the best mix of cognitive abilities and experience. They are the people who should be the CEOs. [There was an article in The Economist about this a few years ago - alas I cannot find it.]

BTW, cognitive decline is measurable starting at age 45 -
January 6, 2012 —Cognitive decline is detectable in persons aged 45 to 49 years and may not uniformly start later, in persons aged approximately 60 years, as previously thought, new research suggests.

The study, using data from the longitudinal Whitehall II cohort study, followed participants aged 45 to 70 years at baseline using 3 cognitive assessments over a period of 10 years. The investigators report that average performance in all cognitive domains except vocabulary, which is known not to be affected by age, declined over the follow-up period in all age groups, including persons aged 45 to 49 years.

Archana Singh-Manoux, MD, with the Hôpital Paul Brousse, Villejuif, France, and colleagues from the University of London, in the United Kingdom, reported their findings online January 5 in the BMJ.

we need to not waste having a young person being a janitor.

Talk about being in a bubble - janitor work is very hard on the body, and a person's body is unlikely to last past 50.

And paralegals are the ones who need to pay the most attention to details and deadlines that the old codgers are likely to miss. I say that as a 63 lawyer BTW.

I take your point about janitors - my main thrust was that young people should be doing the demanding jobs and the elderly should perform more simple tasks. For example: working at a self serve gas station etc.

"And paralegals are the ones who need to pay the most attention to details and deadlines that the old codgers are likely to miss."
Given what you say isn't it a bit peculiar that Lawyers earn so much more. Perhaps it should be the other way round.

And I am aware that a lot of difficult legal analysis is often carried out by young associates which older lawyers then use. However this may not be the most efficient use of resources. A law firm might function more efficiently if younger lawyers in their 20s and 30s occupied senior positions. They are able to make hasty judgement calls based on complex information more easily than an older lawyer.



Agreed. Useless bread-gobblers, they are. Eating up all of our health care dollars.

I can't tell what the left axis is. Can somebody help me out?

The answer is not $400,000. More at

Technicality. If you defined "the lottery" in the question as a taxless, lump-sum prize, I suspect the slope would be similar.

That guy at actuarialopinions is trying to be funny. In a Robin Hansonian way I am tempted to talk of some sort of scientific faux-humor as a means to signalling intellectual superiority.

You need more made-up buzzwords.

Actually I was being serious.

Was it the use of 'faux' that you found to be intellectually challenging?

Technically, the answer is not $400,000. That means $400,000 is the wrong answer.
If you asked me that question, I would ask for more information.
The problem here is that the question is vague. It does not define the lottery as taxless, lump-sum prize. If you asked the question as it is phrased, I would not assume a taxless, lump-sum prize, because lotteries I'm familiar with do not work that way. And I would not answer $400,000.
You can suppose the slope would be similar, but you don't know. I don't know either.
But I know that $400,000 is the wrong answer to the question given the way that virtually all lotteries in the United States operate.

So, unless we're considering an alternative hypothesis of older people doing extra calculations to include taxation and non-lump-sum prizes, rather than simply being unable to do the math to arrive at the answer at which the question is obviously aiming, what is the purpose of raising this objection?

They should ask people "What is 2,000,000 divided by 5?"
Not 'If people divide how much will each of the people get from ?'

They should ask people “What is 2,000,000 divided by 5?” Not ‘If [numberic symbol] people divide [number value written out longhand] how much will each of the people get from [ambiguous financial scheme portrayed as game whose rules may vary from locality to locality]?’

numeric. that is.

Fulano is right. Nothing in the question requires that the pot be divided equally.

Lotteries outside the US often pay out the whole sum.

You speak like "outside the US" was a real place. You may believe that, but the distinctly unfunny dude at actuarialoinions doesn't seem to agree.

Most moronic technicality I have seen yet

Unless the goal is to elicit creative responses, why not just ask "What is $2 million split into 5 equal parts?"

Survey creators are unnecessarily distracting from their supposed goals.

...... because if 90% got it right people wouldn't blog about it?

Good for bloggers but bad for researchers who care about measuring something meaningful.

If your finding devolves into debates as to the many ways in which the questions asked can be interpreted, you're doing it wrong.

I can think of two reasons : (i) They believe that they are also testing whether people apply the math they have learnt into real life situations; in other words, it is not just a math test but a math + math modeling test; (ii) They are "math education" type folks and think that phrasing like "What is $2 million split into 5 equal parts" is too technical. I can't for the life of me understand why in this country (US) they teach addition of fractions by making kids cut a paper circle into 24 disks.

Disks, not sectors?

How do they teach it elsewhere? I really would like to know.

Here in Europe we do not cut paper or anything. Teacher showed us how addition is done and a little bit of explaining and that is.

What European said. In India I did not cut paper either. If you make them cut paper, their attention will be on cutting paper, not on learning addition of fractions. The text book had pictorial illustrations though, and I think that is fine.

I had a college professor who tried the "surprise, we're actually in base-8 math" thing once, with no hint at all ahead of time we were in that.


However bad the question may be, it's the same bad question for everyone. So the *level* may be misleading,( eg a clearer question might put younger respondents closer to 100%), but the *trend* is unaffected.

The older the person is, the more difficult that person has guessing the answer the surveyor wanted. Therefore this graph is clear evidence of some type age-related cognitive decline. I don't care if that cognitive decline is related to social skills or quantitive skills -- it's a decline I would want to avoid.

From a grammatical perspective, $2 million would also be a correct answer.

Maybe I don't want to become that old.

I'm pretty sure, Andreas, that when you are 79, you will want to become 80. Assuming, that is, that you aren't in pain etc.; if you take care of yourself there's no reason why you shouldn't be happy and healthy into your 80s.

Agreed, but if you really feel that way it's one of the things in your life over which you have total control.

What an excellent application of marginality. Kudos. :)

Makes you wonder about the value of economists who premise their theories on rational choice models.

Makes you wander about the value of intrenet trolls who premise their comments on beliefs not touted by anyone.

You're kinda touchy about beliefs that no one holds.

You're kinda touchy about SPINACH

Micke, I am tolerant of you even if your beliefs may not touted by anyone.

Such self introspection and self recognition.

The question, as quoted above the graph is unclear as it doesn't state that the $2m is to be divided equally; a plausible interpretation is that each winner gets $2m.

This was what occurred to my mind too; it is possible that the interpretation of the meaning of total lottery prize changed over time, and that people currently 90 have the "old school" interpretation. However, if that were the case, would one expect a relatively "continuous" curve like this? I would expect the interpretation to change more discretely across age groups.

Yes, that was my thought as well.

And how OLD are you?

I didn't follow the link to read more about this. Do we know what the other answers were? Are they answering $2M? I think knowing the other answers would help us have an idea if they are giving a plausible answer or not.

OTOH, if the answer given is "Get off my lawn, you damn kids!" the answerer should be given extra credit.

Love how Alex's implication is "Government should spend less resources preserving old people's bodies!" The progressive implication would be "Government should spend even more resources preserving old people's minds!"

We don't know how to preserve old people's minds.

That's true, but government does know how to spend money. ;)

Put them (old people's minds that is) in a jar with some pickling solution.

It worked in Futurama

I thought it was, "The government should discount old people's votes."

Wasn't the whole point of government-supplied medical care to "control costs" i.e. more spending = bad according to progressives?

What am I missing? Seems kind of "heads I win, tails you lose" to me. If we spend more on health care, then we need more government interference in order to control costs. If we don't spend more on health care, then we aren't providing an adequate safety net for old people.

Serious question for progressives: What would the circumstances look like such that we would have very good reason to lower spending levels? Or, is every set of circumstances an argument for a spending increase?

@joshua: Alex isn't implying that, you're just projecting.
And as for the "progressive implication" --one would hope that the progressives will eventually realize that "even more resources" don't magically appear from thin air, and that someone will think to do a cost/benefit before getting too carried away with this idea.

(in case anyone discounts this as "if you knew what a family with this goes through"--I'm spending well into six figures per year so my mother in law, with Alzheimers, can stay in her home)

None get $2 million.

The question says the number who get "THE prize" of $2 million.

The is a definite article.

Your elders are smarter than you are, or took an English class.

it doesn't say "get the prize of $2 million"

you've rewritten it to misstate it and then called out someone else for bad reading comprehension

He paraphrased in order to show the questions intent. You're splitting hairs grammatically, in an attempt to be obtuse. The bottom line is the question is a simple story problem that asks "what is 2 million divided by 5 ?" Play with the semantics all you want, after all this is the internet.

if he rephrased it to accurately show the intent, why does he think the answer is $2 million? Of course, that's not what he did, he rephrased it to change the desired answer, and did so after "splitting hairs grammatically"

"THE prize is two million dollars..."

What's the paraphrase?

OK, Careless, if you don't see this interpretation, then let me introduce another ambiguity: there was only one winner.


Because there had to be one person who went to the lottery officials FIRST to claim the prize.

The others followed, but were not FIRST.

Actually, when you do lottery rules or auction rules, you have be specify everything pretty carefully.

I've written on how auction rules determine outcomes, and how one should bid based on what the rules are, and how clear they are. Here's an example: two person reverse action between Company A and Company B for the business of a customer. A and B bid in the first round. The customer says "You have to do better." A continues bidding until he gives up, saying "This is just too low for me to continue." Customer responds: B stopped bidding three rounds ago, and you were bidding against yourself. A's failure was in not negotiating the auction rules to require that bidding ceases when the other person drops out.

Unfortunately, large jackpot lotteries feature parimutual payouts all over the world. But why worry about getting the institutional details right when you can rant about something unrelated instead?

You are relying upon something (parimutual payouts) that is not stated in the rules to fill the gap.

What if you can't fill the gap?

of course I see the ambiguity, I called you out on rephrasing it to turn ambiguity into definite and incorrect meaning. It was already bad writing, but you changed it there to something it had not been.

Careless: Read my lips, as they would have said in the past: the text says: and I quote with emphasis: "THE prize is $2 million...."

when you're talking about the mental decline in seniors, all this stuff is irrelevant when he larger point is that at 50 they can produce the desired answer but at 90 very few can.

Oh, and Careless, it says: "THE prize is $2 million..." emphasis added.

Oh, and Careless, would your answer have been different if the prize were an indivisible $2 million yaucht because if you believe that the word THE has no significance you should be able to substitute one prize for another of equivalent value.

Look, it's clearly worded less than precisely. But isn't that reflective of real world conditions? I think the point of this survey is to emphasize why it is that old folks are so easily suckered by financial scams (and all the real world data shows that they are highly disproportionately likely to do so) -- because, for whatever reason, they've lost the ability to accurately judge the risk/reward ratio. And if they think all 5 winners will get $2 Million, that's pretty strong evidence that they can't judge that r/r ratio.

Heck, if anything the question in the title is *more* clearly worded than the kinds of scams that prey on seniors.


Why can't all five each get $2 million. It didn't say the total was two million to be divided among each winner. It said the winner would get two million.

creative y-axis makes the issue look worse than it is.

Actually it makes the issue look better than it is, at least with respect to the 50 year olds.

Not really, inasmuch is the "issue" is the effects of age on mental facutlies rather than a general lack of numeracy.

When the Powerball prize is large, you will hear people in pubs talk about the cash value of the top prize. They usually discount the nominal value by half, because it is paid out over a number of years, and figure that taxes will take another chunk that they'll never see. So their value of a $2,000,000 prize will be about $600,000. Then divide that by 5 and ....

Remember the question asks how much the winners get ... not how much is allocated to them.

Which is why I would like to see the other options they had for answers. Or most common other answers given if it wasn't multiple choice.

If the modal answer was $2 million each, then I would be thinking that the question was misunderstood. If it was between $2 million and $400,000, I would think innumeracy,. If it were under $400,000 I would seriously consider that the old people are including taxes in their assumptions. And what is the age trend on incorrect questions: Are people more likely to misunderstand the question or more pessimistic about taxes as they get older?

As all lottery prizes in the US I know of are based on the total payout over 20 years, the winning for someone who is 90 can only be $400,000 if the winner lives to age 110, which is actuarially unlikely.

No one will collect $400,000 in cash, but will get the present value of a 20 year annuity paying $20,000 annually with payments going to the beneficiary or the estate.

And then taxes must be withheld for State and Federal with essentially no exceptions (refundable on filing tax returns).

What is scary is the ignorance of those who came of age after Reagan became president and conservatives started selling people on the reality of free lunches in their campaign and policy debates. I think people just want to believe in the tooth fairy, magic, the free market, 10% annual minimum returns for everyone in the stock market, and Social Security and Medicare being private free market institutions so eliminating government won't effect them. In 1980, gambling was either criminal, sinful, or done only at the morally suspect Catholic churches which America was founded on protesting the tyranny of Rome - now government seems to be an arm of Rome and the Pope.

Started out normal, then meandered into the most insane post of the year?

Lou, you should have anticipated that based on it being mulp.

for the record, neither of the two big lotteries are on a 20 year schedule. 26 and 30 now, apparently.

Is this really that surprising? First of all, when I retire I guarantee you the last thing I'll be doing is algebra. Second, we have these fantastic little devices called calculators that perform simple math for us. Problem of old people not being able to do math solved.

From what I see from undergrads, calculators can actually make things worse. They press buttons and an answer comes up and they take it as Gospel. Even if they get .000002 because they entered something in wrong, they'll believe it. They put no thought into whether or not the number that comes out is reasonable. If you point out that their answer is wrong, they throw a stink. "Well, my calculator must be broken" is an excuse I hear way too often.

And because they're so used to calculators, they're even more completely hopeless without them. Even it if I pointed out that it's just 20 divided by 5 with some extra zeroes, I'd wager than no more than 70% even gets a number that starts with 4, much less the right number of zeros.

It's not just undergrads. Remember when Megan McArdle was calculating something with tax cuts and deficits on a per capita basis and got a number off by a factor of 10 or 100 or something and blamed it on the fact that her calculator didn't go to the billions?

Just because Megan McArdle is a moron doesn't mean everyone who uses a calculator makes this kind of mistake


+1. IMO, the big problem is not so much that people cannot accurately perform mental calculations, but rather they lack an instinctual grasp of relative orders of magnitude. This is becoming a major problem on the political level, where -- especially in the US -- the numbers are so beyond what people deal with in their own lives. I think a key part of the deficit/debt problem is that people don't really understand the difference between a million and a billion, or between a billion and a trillion. That is why politicians are able to get away with making claims (or a least implications) that are palpably untrue to anyone who has such an instinctual grasp -- e.g. that the US can solve its fiscal problems by taxing Buffett and other rich people more (on the left), or that the deficit is due to earmarks/foreign aid/NEA grants (on the right), or that middle income earners are overtaxed and get hardly any benefit from government spending (left and right).

And if you're cynical, you might think politicos exploit this. Just to pull some numbers out of the air: We spend 3.4 trillion, we cut 163 million, 163 is bigger than 3.4 so emotionally speaking, that must be a big cut, right?
And if you're not cynical, explain why no one puts it as a 163 million cut from a 3,400,000 million budget, or a .005% cut.

Fair enough about undergrad usage of calculators. However I'm willing to bet that if this question was asked with the use of a calculator a much larger portion of the elderly would answer correctly.

I'm pretty sure that splitting $2M into 5 equal parts doesn't require a knowledge of algebra.

Arithmetic? Too bad it's a subject that has become lowly in the modern American schooling system. I hear "multiplication tables" are frowned upon too as "rote learning".

We'd call it straight division in my household. My 7-year old was forced to memorize the multiplication table up to 12 x 12.

Yeah so was I - didn't do me any fucking good.

The difference is that my 7-year old isn't full of self-pity nor has he made it his mission to sow (sour) grapes across the blogosphere.

You never know - I was a happy kid too but when the disappointments mount one day he might be the next CBBB

The modern schooling system? You realize the chart is talking about what children were being taught ~80 to ~40 years ago, right? Not what they're being taught today. The people most likely to be exposed to "modern schooling" such as "new math" are the ones who do better on the chart than the older ones who were "taught to the tune of the hickory stick."


You're right. Need better evidence for my pet rant.

Commenters are overthinking this. I watched my father through his stages of Alzheimers over many years. When he retired at 62, he was upset if he was $.02 different from the bank when he balanced his checkbook. By 87, he no longer could write numbers accurately. If I had been able to graph the decline, I expect it would look exactly like Alex's graph.

I'd be interested in seeing "what is $2 million divided by 5" charted by age. I suspect it would not match Alex's graph. Still though, it's unfortunate that Alex doesn't have a better example for his overall point, which I think is spot on. It doesn't make sense to keep people alive physically if they are mentally incapacitated. As someone who has dementia and Alzheimers in their family, I hope we can make progress in the next 15 years before it becomes and issue for my father and 45 years before it likely becomes and issue for me.

Well in real life people don't walk up to you and ask "what is 2 million divided by 5?" It's your ability to figure out these kinds of word problems which matters more on a day to day basis.

I could have chosen a better example. Based on the other comments in this thread, the lottery question isn't a good one given all the complications of lottery payouts, which muddles the interpretation of the chart above. Critical thinking matters, its also important to measure it effectively.

And also, just not knowing how lottery payouts work. I mean, if you don't play the lottery, I think it'd be reasonable to imagine that in the case of multiple winners, they each get the full prize or something. So each person gets $2 million.

It's definitely a weird question -- it feels like failure on it could come from a wide variety of failures:

1. Inability to divide 2 million by 5.
2. Understand how lottery works, able to divide 2 million by 5, but unable to parse this problem into "divide 2 million by 5."
3. Do not understand how lottery works.
4. Overthinking the problem and, for example, trying to take into account taxes or 20 year payouts.

I think that we're meant to believe that the slope of the graph (if not, perhaps, the height) is the result of 1 & 2, but it's not clear to me that 3 & 4 don't play a part.

From the paper:

> Finally, we present two measures of practical numeracy. 4e plots naive and control performance in response to the question: If the chance of getting a disease is 10 percent, how many people out of 1,000 would be expected to get the disease? At age 53, 79% answer correctly. By age 90, 50% answer correctly.

Toss in a tiny bit of valuable critical thinking (I agree with Urso), and there goes another 40% of performance, perhaps...

The original presentation had some better (less ambivalent) examples. Kind of bad luck they didn't get picked.

I had no idea the readers of MR skewed so old.

This is ugly data, and one would hope that it not influence policy very much. I had a great-grand-aunt and uncle that lived to 99 & 109 respectively. They were riotously enjoyable company well into their nineties. I doubt they were managing their own finances.

I don't doubt they consumed a ton of Medicare money. What is your stance on Medicare, and the long term budget?

I guess if they are all answering it wrong it's a sign of dementia but I tought it was a trick question in realistic terms, which includes taxation. In NYC the government takes about half the prize money. So 400,000 but you're likely to get 200,000 each.

Yes the interpretation of the lottery prize can be ambiguous, but in a different slide the searchers ask "If the chance of getting a disease is 10 percent, how many people out of 1,000 would be expected to get the disease?” The curve for ""Fraction of people who answer “100”" look very similar as this one.

Yes, the shape of the curve looks very similar, but the decline is from 80% to 50% right rather than 50% to 10%. The decline is still severe, but the conclusion about the mental faculties of 90-year-olds is not nearly as dire and in the results from the first question.

Sorry, the "better" question is even more poorly phrased. If a CHANCE is 10%, how many people out of 1000...? It can be any number in the range from 0 to 1000, most probably somewhere between 80 to 120. Correctly rephrased:
"If the chance of getting a disease is 10 percent, how many people out of 1,000 would be expected to get the disease in the MOST PROBABLE outcome?"
"If every 10th man contracts a disease, how many people out of 1000 will get the disease?"
If roughly half of the population is innumerate, I wonder how low the probability and statistical literacy is.

Huh, yeah. I took “expected” to be intended as in “expected value”.

@alex tabarrok. Unlike you, I would expect cohort effects to be significant. Education is known to affect calculus abilities and the question is well documented. It is unlikely that 90 years old people are as well educated as those who are only 50 over the considered period (20s to 60s). The article should report (if possible), at least the effect of education on the results to appreciate cohort effects. Thus, I would better agree if you stated that no matter cohort effects, the level of the curve is not important, we are interested in its shape. The question of education is however important for who wants to look at the distributive aspects of the policies preconized by David Laibson.

Cohort effects in intelligence testing are significant whenever we do try to measure them; why on earth would we just assume they aren't significant when we aren't looking?

Cohort effects are likely to be important : (Figure 2 tells the story).

What most worries me is the flattish slope at the 50 year mark. Does this hint that even at age 30 we'd have, say, 40% of the population answering incorrectly?

An innumerate 30 year old scares me more than a demented 70 year old.

Well, I guess we can be thankful for the fact that our demented elders have little power and receive a small fraction of total government spending.

I have a different objection. As the question is worded, there could have been 10 winning tickets sold, with 1 of the 5 people holding 6 of them and the other 4 one each.

How many of the 90% of 90-year-olds who didn't answer "400,000" gave any answer at all? Maybe it's in part a graph of the willingness to give any attention to researchers' questions.

There is no way that I would have answered 400,000 for this question. For one thing, suppose that the respondents lived in different states,then they would have different taxes. Suppose the winners had different incomes, then their tax rates might be different even if they lived in the same state.

Perhaps this graph is really a graph of the % of people who have never had to fill out their own tax return? Those who lived longer were probably wealthier originally, and so more likely to be familiar with taxes. I have no idea what the other answers were, perhaps one was <400000. There are just so many ways this is a pointless graph.

I always thought many of the commenters at MR were suffering from dementia. Now I know they are.

The graph is so monotonic at first glance I thought it was some sort of cumulative graph. "The fraction of total respondents who answered correctly above this age"

The others incorrectly responded "GET OFF MY LAWN."

Two words: Flynn effect

Flynn is ~3 IQ points per decade. Can12 IQ points explain this difference?

I was also wondering how we go the point that about half of 55 year olds got the answer wrong. Scarey!

And given this data perhaps in the same way we have age 18 as the minimum age to vote we should have age 70 as the maximum?

A post bound to generate 100+ comments. Personally, I agree with RZ0: the correct answer at any age is "I don't know, because you haven't given me enough information". The graph may in fact indicate that people get sharper and wiser as they get older.

Why did they start at 52?

I'm guessing that the numbers for people younger than 52 show something the authors don't want us to see.


Maybe the older people just understand tax effects better.

Also, I'd like to see this data broken our by ethnicity and region, just for fun.

Older people are mostly white. Are you suggesting whites are particularly likely to be innumerate?

Actually, I was expecting to see whites' scores fall off faster in the areas where lard is still part of the diet. I also expect to see Latinos start out behind whites and move ahead at later ages, Asians to be ahead of the curve everywhere but even more so at the later ages, and blacks to do the opposite of Asians. All of these are for reasons of diet and nutrition, btw.

Interesting, because while I was searching for that census link I came across this beef industry study figuring out what races and age cohorts are more likely to like certain kinds of beef.

"The slide is from David Laibson who has much more on aging and dementia;"

Wow, great citation, Alex! Although to be fair it is more specific than "who has much more somewhere on the internet." Anyway, good to know that the slide appears somewhere in one of the presentations Laibson's done since 2008.

Meanwhile, since I'm actually old enough to remember when current 90 year olds were working adults in their 30s I'm... pretty sure that even in my relatively "high tech" manufacturing town (Eastman Kodak, Mead, lots of defense plants) the rate of high-school graduation was pretty low relative to those of us now in our 50s. Even more to the point, after graduation the frequency with which one might have been called upon to do "story problems" involving fractions of millions would have been pretty low for most contemporary 90-year-olds. Or, for that matter, quite a few 50-year-olds!

That's all pointless speculation, though, since we have no real idea because the slide contains no context for assessing what factors were or weren't accounted for. And the odds of me spending time trying to track down one slide in Laibson's CV in hopes of determining the context is considerably lower than two million divided by five.


Or maybe if you clicked on the link you would see the first presentation listed which just happens to have the same title as the title of this post, The Age of Reason get it?

Actually, I did a lot of digging and found the original reference to this slide. The authors themselves say that the wording was poor on this question. The have a second graph showing the percent that correctly could say how many people out of 1000 would contract an illness for which the probability of contracting is 10%. That graph starts at 80% and follows a similar decreasing pattern. So, perhaps the poor starting performance is due to the poor wording of the question, but the trend with age seems more robust.

A excellent example of the uselessness of powerpoint. There is still a dearth of context.

I liked the comments to this post so much that I created a game you can all play.

The Game is called "Make Bill Rich"

It goes like this:

For just $1, you can bet on a number between 1 and 10. If you guess the correct number, you get the $15 prize.

Send your $1 and bet to this website, addressed to Bill c/o Alex Tabbarok.

NOTE: Before you send the money, ask yourself this question: Will Bill give you $1 if you guess the correct number, or will he give you $1 divided by the number of winners guessing the correct number.

Ooops. This offer is void because it is prohibited by Virginia law and the jurisdiction in which I reside.

My Note comment, not the offer, should reflect the payoff of $15, not $1.

I don't even understand your premise. Your giving out either $1 or $1 divided by the number of winners aren't real outcomes given that you stated the prize is $15.

Secondly, you didn't state that "each correct respondent receives a portion of the prize." This whole thread is more telling regarding a dearth of communication/English skills than it is a lack of math skills.

Read the post: "If 5 people all have the winning numbers in the lottery and the prize is 2 million dollars, how much will each of them get?"

Read my offer: "For just $1, you can bet on a number between 1 and 10. If you guess the correct number, you get the $15 prize." How much will each of the winners get?

On "also raises issues of the value of medical care that maintains the body but not the mind."

What's surprising with this? (apart from the low share of non-elderly getting this wrong). Aging deteriorates the physical as well as mental capabilities. Would anyone be surprised by seeing that the muscle strength of 85yo is lower compared to 50yo?

And, I do not quite see how it "raises the issue" of the value of medical care. Are they less happy with their lives due to lacking numeracy? Quality of Life? WTP for life expectancy improvements? Further, I am not so sure what care maintains the body. Its about slowing down the deteriorating process (for body and mind).

The best group on which to perform this survey would have been the US Senate.

Dead, I see your point and changed the payoo to $15.

It's very satisfying I am sure to critique the question or graph, but what about the point that the post is making? Does anyone really believe that the conclusion is not true? I bet a graph of 100m dash times would look similar. We know that the body deteriorates with age, so why not the brain?

I wonder if this result is related to the well known result that happiness increases with age (once you are over 50)? We hear from the HBD folks that lower intelligence usually means poorer future time orientation, older folks are living in the now baby!

You don't have to find errors in reasoning to be associated with age.

Question: A bat and a ball costs $1.10 in total. The bat costs a $1.00 more than the ball. Does the ball cost 10 cents? True or false.

Question: If it takes 5 machines to make 5 widgets, how long would it take to 100 machines to produce 100 widgets. Does it take one minute. True or false.

Answers: A majority of Graduate students failed to answer the questions correctly.

Answers: 5 cents; 5 minutes.

"You don't have to find errors in reasoning to be associated with age".

Just so we are clear, the study did find errors in reasoning to be associated with age.

Does your answer fall within the category as a wrong answer associated with age, or as a wrong answer. It's hard to tell if the underlying question raised in this post is deficient. But, it is also the case that logic errors occur at all ages, which is my point that errors in thinking occur at all ages.

I think.

Your second question needs some educated mind-reading to answer correctly. :)

I wonder what percentage of people can also state the assumptions in the question (e.g. “Prize is split equally” and “No other people have winning numbers”).

Also would be more appropriate to not write the answer on the graph.

And the aforementioned graphed range — it really should start at 0 (not 0.1), preferably end at 1 and would be useful to have some data for younger ages on it.

Like many pointed out I would disregard the study completely.

I am in my 30s and I doubted too the answer:
1) Is 2 million for each or divided among the 5.
2) Do they 5 have equal rights to that among (equal participation weight)
3) Has the 2 million, depending in the place, 0, 30, 40%, etc deducted in Taxes already or not?

Older people usually are more skeptical so I would not doubt those 3 factors weighted more as age progresses. The question should have been simply divide 2 million among 5; and even that would not account for age deterioration but maybe reflecting past generational education differences.

Also we should account the time they need to give an answer; it is not the same standing up in a mall than in a room with plenty of time.

So as you get older your over simplistic models of the world become less over-simplified. ;-)
No one gets the full prize money for the lottery due to taxes on winnings, and various other things.

Hard to tell if represents declining individual abilities or an increasing improvement in education over time.

50 year olds were educated in the 1960's and 70's and had the benefit of civil rights and Great Society education reforms, 70 year olds may have seen benefits from the GI bill, 90 year olds grew up in the 20's and 30's when progressive education was an experimental idea that very few could take advantage of.

I've got to say, the comment about medical care for the body despite the mind really hit home for me. I'm a surgery resident, and one of our patients has been in the hospital for most of the last 5 months. Her body is mostly toast but we can make her live a lot longer with dialysis, etc. Her mind is absolutely toast.

We pour billions of dollars in health care costs into saving people whose minds are shot all of the time. I just don't believe that bodies without minds are worth much.

Maybe, just maybe, older people do not give a ---- about long division.

Well, these are Americans. And as we in the civilised world know, 68% of Americans believe that angels are real and affect their daily lives (Pew Poll, 2007), and over 90% claim there's an invisible Sky Wizard who loved foreskin so much that he traded mastery of the world in exchange for an endless supply (having created a hundred quintillion stars in one day... why couldn't he make his OWN foreskin?).

Anyhow... basic point: expecting mental dexterity in basic arithmetic seems a tad 'cart before the horse' when your sampling universe includes a vast number of people who can't think past a level that most western 4 year olds would mock.

Being a lawyer I’m somewhat biased, but imo the question is neither 1) a mathematical problem nor 2) a problem that it’s up to you to solve.

1) Since the question does not specify that the prize money is shared among the winners, it’s entirely possible that each winner is entitled to the full 2m.

Whether or not the money should be divided is a legal problem and depends on a court’s interpretation of the legal contract you entered into when buying a lottery ticket. This may not be the “desired” answer, but since the desired answer automatically forfeits 1.6 million, it’s hardly desirable. Thus if anything, the answers get more accurate with age.

2) Someone asked what would happen if the prize was, for instance, a yacht. It isn’t divisible, but that isn’t your problem. That is a problem for whoever wrote a contract promising one non-divisible thing to several people. I don’t recall what the name of the legal areas that deals with these questions in English, but suffice to say it’s not a new problem and all legal systems have solutions.

I would have guessed 200,000 based on 50% gov confiscation, Then 100,000 after angry wife takes her share.

Point to a real world instance where five winners of the same lottery took home the same amount.

Otherwise, you're just playing the "Boomers Suck" card.

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