Economic impact of mathematics?

Tim Gowers wrote in The Importance of Mathematics:

If you were to work out what mathematical research has cost the world in the last 100 years, and then work out what the world has gained, in crude economic terms, then you would discover that the world has received an extraordinary return on a very small investment.

I don't doubt this. But has anyone actually tried to do this? (And would the numbers even be meaningful?)

When 2+2 = 5

From "you suck at craigslist:" 2+2 = 5. People selling things on craigslist who want $x for one of some item and more than $2x for two.

I'm pretty sure I once saw somebody selling T-shirts on the boardwalk in Atlantic City for "$3, 3 for $10." Or it might have been "$4, 2 for $10" or "$2, 4 for $10". In any case, it didn't make sense.

I'm sure there are cases where this sort of pricing structure actually makes sense, but I can't think of anybody. (I also wouldn't be surprised to learn that economists have a name for it.) Anyone want to try?

The third derivative of the employment rate is positive

The third derivative of the number of people employed in the United States is positive. (From 538.)

Nate Silver puts it as "the second derivative has improved", but let's face it, this is really a statement about the third derivative. Compare Nixon's 1972 statement that the rate of increase of inflation was decreasing, which Hugo Rossi pointed out in the Notices was a statement about the third derivative. (I seem to recall John Allen Paulos pointing this out in one of his books, but I don't recall which book and therefore can't date it relative to Rossi's letter in the Notices.)

Housing prices drop 13 percent -- what does this mean?

The New York Times reports on bad housing news:

The median price of a home plunged 13 percent from October to November, to $181,300 from $208,000 a year ago. That was the lowest price since February 2004.

They mean that house prices have gone down 13 percent in a year, i. e. from November 2007 to November 2008. That's what the National Association of Realtors press release says.

But one sees this pretty often -- the confusion between monthly declines and annual declines. And sometimes a 1% decline in a month might be reported as a "12% per year" decline -- but then the "per year" gets dropped, the statement "prices of X dropped 12% this month" is made, and those who aren't familiar with how people who care about the price of X report their numbers get confused.

Don't get me wrong -- a drop of 13% in a year is still a big deal. But a drop of 13% in a month would be a much bigger deal.

Will McCain raise taxes?

Greg Mankiw calculates the probability that McCain will raise taxes, using data from the Intrade prediction markets, which have contracts for who will be elected president and for what tax rates will be in the future. It seems pretty likely.

Of course, the error on these markets is pretty high, and the calculation requires subtraction, which just amplifies these errors. But it's an interesting thought. (And I have to admit I've played around with trying to extract conditional probabilities from prediction markets myself.)

Lower speed limits, part two

One thing people complain about in regards to slower speed limits, which I wrote about earlier today, is that when speed limits are lower it takes longer to get places. This is, of course, true. But on the other hand you use less fuel.

From Wikipedia on fuel economy in automobiles: "The power to overcome air resistance increases roughly with the cube of the speed, and thus the energy required per unit distance is roughly proportional to the square of speed." Furthermore, this is the dominant factor for large velocity.

So let's say your fuel usage, measured in fuel used per unit of distance (say, gallons per mile), at velocity v, is kv². (k is some constant that depends on the car. A typical value of k, for a car using 0.05 gallons per mile at 60 mph, is 0.000014.) Let's say you value your time at a rate c -- measured in, say, dollars per hour, and the price of fuel is p.

Then for a journey of length d, you'll spend dpkv² in fuel, and cd/^v in time. Your total cost is

and differentiating and setting f'(v) = 0, the optimal speed is (c/2pk)^1/3. The cost of the journey at this speed is

So according to this model, if you value your time more you should go faster; not surprisingly your value of time c and the price of fuel p show up only as c/p -- effectively, your value of time measured in terms of fuel.

Also, the optimal speed doesn't go down that slowly as p increases -- it only goes as p^-1/3. But a doubling in gas prices still leads to a 20 percent reduction in optimal speed -- perhaps roughly in line with what people are suggesting. Taking c = 10, p = 4.05, k = 0.000014 gives an optimal speed of 45 miles per hour, although given the crudeness of this model (I've assumed that all the fuel is used to fight air resistance) I'd take that with a grain of salt, and I won't even touch the fact that different people place different values on their time and get different fuel economy. We can't just let everyone drive at their optimal speed.

Besides, part of the whole point of this is that if we use less fuel, demand for fuel will drop significantly below supply and oil prices will go down. So to forecast the effects of a lower speed limit I'd have to factor in that gasoline could get cheaper -- and let's face it, I can't predict the workings of the oil market.

Five miles an hour = 30 cents a gallon?

"Every five miles an hour faster costs you an extra 30 cents a gallon." From yesterday's New York Times, among others. This is often mentioned in reference to bringing back the national 55 mile per hour speed limit.

What does this even mean? I assume it means that it takes, say, seven percent more gasoline per mile to drive 65 mph than to drive 60 mph. (30 cents is around seven percent of the current average gasoline price, $4.10 or so per gallon.) Why not just say that? This also has the advantage that when gas prices change, the fact doesn't become outdated.

Although as many people point out, the lower speed limit is a hard sell, in part because of the value of time. If you're about to drive 65 miles at 65 mph, it'll take you an hour; say you get 20 miles per gallon, so that uses 3.25 gallons of gasoline. Slowing to 60 mph, it takes five minutes longer, but saves seven percent of that gasoline, or 0.23 gallons -- perhaps $1 worth. So if you value an hour at more than $12 (more generally, at more than three gallons of gasoline), you should drive faster! Of course I've committed the twin fallacies of "everything is linear" and a bunch of sloppy arithmetic, and I've ignored that different cars get different gas mileage, but the order of magnitude is right -- and it's clear to me some people value their time at more than this and some at less. And a better analysis would take into account the probability of getting in accidents, speeding tickets, etc. (I'm mostly pointing this out because otherwise some of you will.)

Oh, and on a related note, people will do things for $100 worth of gas that they wouldn't do for $100 worth of money.

Why medians are dangerous

Greg Mankiw provides a graph of the salaries of newly minted lawyers, originally from Empirical Legal Studies.

There are two peaks, one centered at about $45,000 and one centered at about $145,000. The higher peak corresponds to people working for Big Law Firms; the lower to people working for nonprofits, the government, etc.

The median is reported at $62,000, just to the right of the first peak, since the first peak contains slightly more people. But one gets the impression that if a few more people were to shift from the left peak to the right peak, the median would jump drastically upwards. We usually hear that it's better to look at the median than the mean when looking at distributions of incomes, house prices, etc. because these distributions are heavily skewed towards the right. But even that starts to break down when the distribution is bimodal.

When it comes to oil, addition is hard

Have we underestimated total oil reserves?, from New Scientist (and, it appears, every other source in the British-speaking world).

Richard Pike, of the Royal Society of Chemistry and previously of the oil industry, points out that it's generally the convention in the oil industry to give, as a one-number estimate for the output of a particular oil, the 10th percentile of the distribution that they expect. This is an inherently conservative estimate. That's fine -- but then when they combine the estimates from every oil source they do it by just adding those together. If I'm understanding this correctly, the estimates that are out there of how much oil there is correspond to what happens if every oil well, etc. performs only at the 10th percentile of its expected distribution -- which just isn't going to happen. The 10th percentile is called the "proven reserves", the 50th "proven plus probable reserves".

I don't know what the underlying distributions are, but it seems like they generally report the 10th, 50th, and 90th percentiles of the underlying distribution -- so says Pike at this friction.tv video. They add these distributions together correctly internally but Pike claims that governments don't want to think about the probabilistic logic. Pike then claims that security analysts use the resulting very pessimistic estimates; perhaps the current run-up in prices is a result of this, although I'm not sure if he'd say this. (It sounds like it's not a secret that this is the way things is done, and perhaps the analysts know this.)

Mathematically, the idea is simple. Let's say that a certain oil field is expected to produce 4 megabarrels, and the production is normally distributed with standard deviation 1 megabarrels. The tenth percentile of a normal distribution is 1.28 standard deviations below its average, so the "proven reserves" of this field would be 2.72 megabarrels, and the "proven plus probable" 4 megabarrels.

But now say we have four such fields. The oil industry's techniques would say that those fields together have "proven reserves" of 2.72 million times four, or 10.88 megabarrels. But for uncorrelated distributions -- and I'm going to assume that the distributions here are uncorrelated -- the variances add. The variance of the distribution for one field is 1 megabarrel², so for four fields it's 4 megabarrel²; the standard deviation is the square root of this, 2 megabarrels. The mean is still 16 megabarrels, but the 10th percentile is now 13.44 megabarrels. The chances of getting as low as 10.88 megabarrels are about one half of one percent; this is what Pike means when he calls this a pessimistic estimate. And of course with more fields the pessimism becomes more extreme.

The first reference I can find to this is a May 2006 press release of the Royal Society of Chemistry, referring to a June 2006 article by Pike in Petroleum Review but it seems to have swept through the British blog world in the last few days after the Times of London made a quick reference to it in an article on North Sea oil, as this press release of the RSC mentions. (For some reason the Times article refers to "this month"'s issue of Petroleum Review, which I can't seem to see online even though Penn's library claims to have an electronic version.)

If this is true (I hesitate to think it is, just because it seems surprising that people wouldn't know this!) then it's good news and bad news. Good news because it means we're not going to run out of oil as soon as we think, which means less economic shock. But it also means more carbon for us to spew into the air before we finally run out.

I encourage you to not read the comments in most places where this has been posted, because it's basically people just ranting about global warming and saying either "we've reached peak oil, anybody who says we haven't is a poopyhead" or "we haven't reached peak oil, anybody who says we have is a poopyhead".

The moral here: probabilistic forecasting is tricky. See also Nate Silver's appearance on cnn.com regarding polling for the presidential election, which is totally irrelevant here except that it also goes to show how a lot of people don't know how to aggregate probabilistic data.

Malthus is overrated

Malthus, the false prophet, from The Economist, and Costs of Living, a review of Common Wealth: Economics for a Crowded Planet by Jeffrey Sachs.


Sachs' book is about how to avert global economic catastrophe, which is obviously something worth worrying about. Basically, he seems to be saying that global cooperation is necessary, because "we're all in this together". He also argues, it seems, that population growth is bad; I found the article from Greg Mankiw's blog. Mankiw quotes the end of the review:

In an age when we don’t need to have lots of children to work the fields, or to compensate for high infant mortality, Sachs argues that it’s both economically rational — and crucial for a future of sustainable growth — for people to reproduce at a rate close to 2.1 children per family. In his acknowledgments, Sachs thanks his three children.

But Mankiw pointed out in 1998 that it's not necessarily true that population growth is bad. It seems like a lot of people like to quote Thomas Malthus on this, who said that population grows exponentially with time (true) but that food production ability only grows linearly. As far as I know Malthus had no reasonable argument for this; people talk about the Malthusian catastrophe. But if you actually look at the relevant section in An Essay on the Principle of Population, in chapter 1 Malthus just postulates these growth rates. In Chapter 2 he offers as justification for this basically that there's not enough room in England for the farms to feed an exponentially growing population. But there's not room for an exponentially growing population itself either!

It would be one thing if, say, farms fell from the sky at a constant rate. But farms are made by people; thus if people grow exponentially so will farms.

Now, I'm not saying that we aren't running out of room. It's obvious that some land is better for farming than other land, and people will tend to farm the better land first; as time goes on we will be obliged to farm more and more of the inferior land, therefore decreasing the amount of food grown per person.

But Malthus was writing in 1798. He assumes that the population is producing just enough food for itself in his time, and then goes on to say:

In two centuries and a quarter, the population would be to the means of subsistence as 512 to 10.

Two centuries and a quarter is 2023; roughly speaking, now. We clearly have much more than two percent of the food we need.

Of course, it's possible -- as is pointed out in every introductory computer science class -- that polynomial growth can outstrip exponential growth over some short time, and we're not in the limiting regime yet. But I don't think anybody is seriously saying this. And anyway, Malthus made the stronger claim that food production is growing linearly.

(Does Malthus get more sophisticated than this? I'm just cherry-picking, but skimming his work it seems to continue in roughly the same vein.)

Now, this is an obvious criticism -- but somehow it rarely gets pointed out. Mankiw pointed it out, though -- sure, people use resources. But they also create resources. People will eat food. But they will also figure out how to grow more food. That is, if we don't just disintegrate into a society of virtual reality addicts first.

edited, 11:08 am: Also from the NYT, Deaths are outpacing births in the Pittsburgh metropolitan area. I'm not sure how meaningful this is on a national level, because people move around.

Well-intentioned money advice

Suze Orman, "internationally acclaimed personal finance expert" (actually the title of her web page!),, said yesterday on myphl17 News At Ten something like: "you are not to spend your economic stimulus check. you must save it." (Don't ask me why I ever watch this newscast. It consists of recycled press releases, news that someone got shot and somebody's house burned down, and sports scores. The only score I care about is the Phillies and I usually know how that turned out anyway.)

Anyway, Orman's advice seemed to be based on the idea that because the economy as an aggregate is doing poorly, we all must be suffering. There are surely some people who have had a very good year and don't need the six hundred bucks. And there are surely some people who have had a very bad year and for whom six hundred bucks is just a drop in the bucket.

I'll call this the "distributional fallacy" (does it have another name) -- assuming that any individual must be representative of some sample from which they're drawn. Not a horrible assumption in the absence of other information -- but I know more about my financial situation than someone appearing on my television!

But "if times have been bad and you don't have money saved up, you should save the money -- and maybe you should save it even if things have been good for you, because they might turn bad" doesn't have the same ring to it.

I'm not arguing that people shouldn't save their money, because life has a way of causing people trouble. But to assume that everybody is going through hard times is kind of short-sighted. Then again, if you tell people "some people should save their money", that American instinct to consume will kick in and people will assume that "some people" doesn't include them.

(For the record, I will be saving my economic stimulus check. I think. It's hard to say, because money's fungible, and I stand to have negative cash flow this summer because I won't be teaching like I have the last two summers. So it'll go into savings, but then I'll spend "it" later. Money is money, it all mixes together. It's a scalar, not some sort of crazy vector in a non-Euclidean space as some people would probably like you to think.)

Prediction markets aren't perfect

As of right now, intrade.com reports a 7.4 percent probability that Hillary Clinton will get the Democratic nomination for President -- and an 8.0 percent probability that she will be the next President.

That seems a bit unlikely to me.

Intrade isn't an efficient market; there are tons of arbitrage opportunities like this. (Some are more subtle.)

Although if you want to get technical, buying Clinton getting the Democratic nomination (at 74 cents for a contract that pays $10) and selling Clinton winning the '08 election (at 80 cents) isn't quite risk-free -- there's a nonzero chance that she might want so badly to be President that she'd run as an independent in order to do so. And it's my understanding that short selling (which would be necessary to pull this off) isn't possible on Intrade anyway.

Breaking down inflation

An interesting informational graphic: All of Inflation's Little Parts, from Saturday's New York Times. This is a graphic that breaks up the many parts of the standard "basket" that is used by the US government to compute the inflation rate.

Of course, nobody is exactly average. To take an example dear to my heart, 0.1% of this basket is "books". But how many people do you think spend exactly that portion of their income on books? My instinct is that the distribution of "percentage of income spent on books" (and a lot of the other small items) has a long right tail. Plenty of people I know (who are of course not a uniform sample) spend, say, two or three percent of their income or more on books -- and it's my understanding that the book industry relies quite a bit on people like us.

Somewhat more seriously, 23.9% of the basket is "owner's equivalent rent" (what homeowners would pay if they were renting their homes) and 5.8% is actual rent. That means that a typical household making, say, $50,000 a year spends about $1,000 monthly on the house they own (or would, if we weren't having a housing bubble), and $250 monthly on rent. But it's very hard to imagine a household that actually does that! The mean and the mode, in distributions like this, are very different. It would be quite surprising, I think, to find someone whose spending breaks down exactly as in the graphic.

The strange mathematics of tipping

Do you tip less in a tough economy?, from Queercents.com, which is apparently a queer personal finance site.

It seems that at some restaurants, the average tip has gone down from the old 15 to 20 percent neighborhood to more like 12 percent. The author of the post comments that this is a "3 to 7 percent" cut in pay.

First, 20-12 is 8, not 7.

Second, that's all wrong! Let's say that before the US economy went all funny, servers made $2.50 an hour in pay, and $7.50 in tips. ($2.50 is somewhere near the federal minimum wage for servers; $7.50 was chosen so that the figures would add up to $10.) Let's say that the $7.50 was back when tips were an average of 18%. If tips average 12% now, then only two-thirds as much tip income will come in -- so $5 an hour.

So our hypothetical server now averages $7.50 an hour instead of $10, a twenty-five percent cut. There's a big difference there. I would not be happy if my income were cut by "3 to 7 percent", but I wouldn't have to make huge changes in my lifestyle. But if my income were cut by 25 percent, there are quite a few things I'd have to cut back on. I suspect this is true for many people.

For my international readers who are not used to the silly system we have here in the US: restaurants typically pay their servers between $2 and $3 an hour, and the rest of their income comes in tips; this is as opposed to the more civilized system I understand you have in much of the rest of the world, in which tipping is reserved for extraordinary service and restaurant owners actually pay their staff a decent wage.

(Thanks to dan for pointing me to this.)

Confusing coffee pricing continued

Last week I wrote about confusing coffee pricing: Wawa, a Philadelphia-area convenience store chain, charges $1.25 for 32 ounces of coffee and $2.99 for 64 ounces. $2.99 is more than twice $1.25. Various commenters pointed out other counterintuitive pricing (train or airline fares that don't obey the triangle inequality, for example). Paul Soldera pointed out in a comment that the reason for this may just be that there aren't that many mathematicians out there, and $3 for 64 ounces of coffee sounds like a bargain to most people.

Paul Soldera may be right -- but I discovered another candidate explanation for this pricing today. Namely, I took a closer look at a sign (at a different Wawa from the one I normally go to), and it said that the 64-ounce "includes supplies". In other words, they're not selling this as a giant cup of coffee for one person to drink, but as something from which you can pour multiple cups for multiple people. Thus, they provide the cups, and perhaps other coffee paraphernalia as well.

Confusing coffee pricing

Here in the Philadelphia area, we have an oddly-named chain of convenience stores named Wawa.

At Wawa, you can buy coffee for the following prices: $1.09, $1.19, $1.29, $1.39 for 12, 16, 20, 24 ounces respectively. This makes sense -- basically you pay $.79 for wandering around in their store taking up space and such, and then 10 cents for each four ounces of coffee.

However, things get weird if you bring your own cup (I'm talking about the "travel mug" sort here, not a paper cup). Then 12, 16, 20, 24 ounces cost $0.85, $0.95, $1.05, or $1.15 -- so far, so good. You save twenty-four cents by bringing your own cup.

32 ounces, in your own cup, is $1.25. So now they're really starting to reward you for buying in bulk -- another ten cents gets you eight more ounces.

But then guess what happens? 64 ounces costs $2.99. That,s right -- I can fill two 32-ounce cups for $2.50, but filling one 64-ounce cup will cost $2.99. If you extrapolate the linear trend from 12, 16, 20, and 24 ounces, 64 ounces should cost $2.15. If I had a sixty-four-ounce travel mug, I'd go in there, fill it up, and try to get it filled for $2.50 just to see how the cashiers explained it.

Perhaps they're trying to say that you really just shouldn't be drinking that much coffee. I'd have to agree -- and I'm a mathematician.

Another argument is that perhaps they are attempting to discourage people from taking that much coffee because then there's less coffee for the people after them, and people won't be happy if the store runs out of coffee. This may be true -- it seems a bit doubtful, though, since a typical Wawa store might have a dozen or so pots of coffee at once, each holding 64 ounces or so.

Hyperbolic discounting

Greg at The Everything Seminar posts about "hyperbolic discounting". Roughly speaking, theoretically one should value a payoff of an amount P of money at time t in the future with the same value as Pe^-rt today, where r is the inflation rate. But people actually seem to value P at that time in the future like they value P/(1+ct) today for some constant c. People probably have a decent sense of what the inflation rate is; thus Pe^-rt and P/(1+ct) should agree locally, so I'd guess c = r. (Set the two expressions equal to each other, so e^rt = 1 + ct, and take the first two terms of the Taylor series.)

Greg writes:

I think something like a logarithmic measure on actual time might give the hyperbolic discounting model.

That's true. Let's say we live at time 0; the correct (exponential discounting) value of a payoff of 1 at time t is e^-rt. The value of a payoff of 1 at time T under hyperbolic discounting is 1/(1+rT). Setting these equal, we get

.
Solving for each variable in terms of the other,

So roughly speaking, from looking at the first equation, the discounting that people actually use instinctively is obtained by taking the logarithm of the time T they're discounting over (up to some scaling, which really just sets the units of time), and then applying the correct (exponential) model. This reminds me of a logarithmic timeline, but in reverse. People see the period from, say, 16 to 32 years ago as being as long as the period from 32 to 64 years ago. This is also why I don't believe in a technological singularity even though I'd like to; the arguments often seem to be based on "look! lots has changed in the past hundred years, more than changed in the hundred years before that!" but our memories of "change" are somewhat selective.

An implication of the St. Petersburg paradox

Many of you are probably familiar with the St. Petersburg paradox. This is the following paradox. I offer to play the following game with you: I will flip a fair coin repeatedly until it comes up heads. If the first time it comes up heads is on the nth toss, I will pay you 2ⁿ dollars. How much are you willing to play this game?

Well, the probability that the coin comes up heads on the first toss is 1/2; in this case you get 2 dollars. The probability that it comes up heads for the first time on the second toss is 1/4; in this case you get 4 dollars. In general, the probability that the coin comes up heads for the first time on the nth toss is 1/2ⁿ, and in this case you get 2ⁿ dollars. So your expected winnings are

2(1/2) + 4(1/4) + 8(1/8) + 16(1/16) + ...

and each term here is 1; the series diverges. So you should be willing to pay an infinite amount of money to play this game. Yet you're not. (If you are, let me know. I'd like to have an infinite amount of money. Notice that you will only win a finite amount of money playing this game, so even after I pay you I will still have an infinite amount of money.)

Yet you're not. You may suspect that this is because you know the person you're betting against doesn't have an infinite amount of money, so your expected winnings don't come from actually summing the whole infinite series. For example, let's say Bill Gates is willing to play this game with you; let's say his net worth is 2³⁶ dollars. Then your expected winnings are

2(1/2) + 4(1/4) + 8(1/8) + ... + 2³⁵(1/2³⁵) + 2³⁶(1/2³⁶) + 2³⁶(1/2³⁷) + ...

where the first thirty-six terms are all 1; then what follows is 1/2, 1/4, 1/8, and so on. So you should be willing to spend $37 to play this game if the Gates fortune is backing it.

Of course, you might argue -- and a lot of economists have -- that $2n is not worth twice as much to you as $n. The usual assumption here is that the utility of $n is something like log₂(n) "utils" (I'm not sure how they handle the problem of the units here), and that people play to maximize their expected number of utils, not their expected number of dollars. Then the "expected value" of the previous game, in utils, is
1(1/2) + 2(1/4) + 3(1/8) + ... = 2
and you should be willing to pay that amount of money which is worth 2 utils to you, namely $4.
But I can construct a similar paradox. If the coin comes up heads on the first toss, I pay you $2. If it comes up heads on the second toss, I pay you $4. If it comes up heads on the third toss, I pay you $16. If it comes up heads on the fourth toss, I pay you $256, and so on... then you receive 1 util if the coin comes up heads on the first toss, 2 on the second toss, 4 on the third toss, 8 on the fourth toss, and so on. So you should still be willing to pay infinitely much to pay this game. And in general one can construct such a payoff sequence for any unbounded utility function.

The somewhat counterintuitive resolution that I heard for this recently is that utility functions must be bounded. So say $1 gives me a certain amount of utility. Then in order to make it impossible to construct a St.-Petersburg-style wager, for which I would be willing to pay an infinite amount of money, there must be some K such that any amount of money gives me at most K times as much utility as $1. I'm not sure I believe this, either... it just goes to show that sometimes expected value might not be the way to go.

More men at the top, and at the bottom.

As has been documented by a lot of people, it seems that a lot of psychological traits have the following properties:

• men and women have approximately the same average for this trait, and


• both genders have an approximately normal distribution for this trait, but


• the distribution of men's values for this trait has a larger standard deviation than the women's values

This has the effect that men are overrepresented at both extremes. The canonical example is skill in science or mathematics; it's been claimed that women and men are on average equally good at mathematics, but most of the best mathematicians are male. This isn't a contradiction, because most of the worst mathematicians are male but we don't notice it. (It actually doesn't matter that the averages are the same; even if men were on average worse than women at math, if they had a larger standard deviation then they'd predominate at the higher levels.)

The article Is There Anything Good About Men?, which was an invited address by Roy Baumeister at the American Psychological Association, addresses this. This is thought to arise from the fact that men can have more offspring than women.

Let's say that the X-ability of women is normally distributed with mean 0 and standard deviation 1, and the X-ability of men is normally distributed with mean 0 and standard deviation σ. Then the probability density function for the mathematical skill of women is



and that for men is



If we look at the ratio f(z)/g(z), this is the ratio of women to men at skill level z. It's



and this equals 1 when

.

When z is closer to zero than this, women will predominate; when z is larger, men will predominate. It turns out that



and since σ probably isn't much larger than 1, men will predominate at more than about one standard deviation from the mean and women at less than one standard deviation from the mean. Furthermore, we have f(0)/g(0) = σ; again, since σ isn't that much greater than 1, the predominance of women over men at the center of the overall distribution is difficult to see.

Yet if σ = 1.1 -- meaning that men's skill have a standard deviation 1.1 times that of women -- then g(3)/f(3) = 1.99, so men will be twice as common as women at z=3 (which corresponds to 3 standard deviations above the mean for women, and 2.73 for men). (The same is true at three standard deviations below the mean.) At z=4, men are overrepresented by a factor of 3.6, and at z=5, by a factor of eight.

Another thing that occurred to me is the economic ramifications of this difference. It's well-known that there are more obscenely rich men than obscenely rich women. It seems to me that economic ability -- i. e. the ability to earn money -- could be proportional to, say, the exponential of (some constant times general intelligence); so for every ten IQ points you gain, your income goes up by 30%. (I made up these numbers.) This would mean that economic ability is lognormally distributed (the name is a bit counterintuitive, if you don't know it, but it means that the logarithm of economic ability is normally distributed). But the mean of a lognormally distributed variable is e^μ+σ2/2, where μ and σ are the mean and standard deviation of the variable's logarithm. So if intelligence is normally distributed in both canonical genders, but men are more spread out than women in intelligence, then the mean of men's earning potential will be greater than that of women. I'm not saying that earning potential is directly tied to general intelligence (I know plenty of people who are smart but not rich) but it wouldn't surprise me to learn that earning potential is lognormally distributed and that something like what I've outlined here is at work.

How much cash should you carry?

An interesting question from Marginal Revolution: how much cash should you carry?

Greg Mankiw has also commented on this; apparently there's a standard model called the "Baumol-Tobin model" that answers this question.

There are two factors in play here that we need to think about. Any cash you're carrying can't earn interest, so you forgo interest by carrying more cash. But it takes time to go to the ATM, and your time has some value; you use more time in getting money if you habitually carry less cash. (If you also have to pay ATM fees, they can be folded into this.)

I suspect that most people, in most situations, manage their cash in the following simple way -- when the amount of cash they have is less than c dollars, they go to the ATM and withdraw C dollars. (For most people, c is probably on the order of the amount of cash they spend in the average day, because they pass by a convenient ATM once a day or so.) In my case, c = 20 and C = 60; that is, when I have less than $20 I stop at an ATM and withdraw $60. The times I don't do this are when I know that I'll have large cash expenditures; for example, last time I moved I withdrew a few hundred dollars in cash because I knew my movers only took cash. The average amount of money I have in my wallet is something like C/2 + c, since it is about equally likely to be anywhere between c and c+C; I'm going to make the simplifying assumption that c is much smaller than C, so we'll call this C/2.

So how much interest do I forgo, annually, by carrying this amount of cash? That's easy; it's Cr/2, where r is the interest rate. (My ATM card is linked to a checking account which gets essentially zero interest.)

Now, let's say each trip to the ATM costs me an amount a, measured in dollars. This is a combination of the time it takes me to get to the ATM, valued at whatever amount I value my time at, and any ATM fees I might pay. If the amount I spend annually in cash is s, then I'll have to make s/C trips to the ATM during the year. The quantity we want to minimize is the sum of the amount of interest I forgo by carrying cash and the time value of my trips to the ATM, which is

f(C) = as/C + Cr/2

To minimize this, we take its derivative; f'(C) = -as/C² + r/2. Setting f'(C) = 0 and solving for C, we see that each time I go to the ATM I should withdraw (2as/r)^1/2. Note that the dimensions work -- a has units of dollars, as does s, and r is a pure number, so 2as/r has units of square dollars.

Off the top of my head, I have s equal to about $4,000 (I withdraw $60 slightly more than once a week), a about $1 (I'm just guessing here, but the "good" ATM is a bit out of my way), and r about 0.12% (it's a checking account -- those don't have interest any more). Plugging in, I get C = $2,581; I should be withdrawing that amount of money every eight months or so. Even if I take r = 4%, which I could get from a high-yield savings account, I get C = $447, which would correspond to a withdrawal in that amount about every six weeks.

Why don't I withdraw that much money? I think it's because I wouldn't feel comfortable carrying it in my wallet; there are psychological factors that this model doesn't take into account, as people point out. Also, I suspect that if I were carrying a wad of cash like that I'd feel richer, which would lead me to spend more money, which would make me actually poorer. Plus, carrying large amounts of cash probably increases my probability of getting robbed...