Summers Vindicated (again)
For the past week or so the newspapers have been trumpeting a new study showing no difference in average math ability between males and females. Few people who have looked at the data thought that there were big differences in average ability but many media reports also said that the study showed no differences in high ability.
The LA Times, for example, wrote:
The study also undermined the assumption — infamously espoused by former Harvard University President Lawrence H. Summers in 2005 — that boys are more likely than girls to be math geniuses.
Scientific American said:
So the team checked out the most gifted children. Again, no difference. From any angle, girls measured up to boys. Still, there’s a lack of women in the highest levels of professional math, engineering and physics. Some have said that’s because of an innate difference in math ability. But the new research shows that that explanation just doesn’t add up.
The Chronicle of Higher Education said:
The research team also studied if there were gender discrepancies at the highest levels of mathematical ability and how well boys and girls resolved complex problems. Again they found no significant differences.
All of these reports and many more like them are false. In fact, consistent with many earlier studies (JSTOR), what this study found was that the ratio of male to female variance in ability was positive and significant, in other words we can expect that there will be more math geniuses and more dullards, among males than among females. I quote from the study (VR is variance ratio):
Greater male variance is indicated by VR > 1.0. All VRs, by state and grade, are >1.0 [range 1.11 to 1.21].
Notice that the greater male variance is observable in the earliest data, grade 2. (In addition, higher male VRS have been noted for over a century). Now the study authors clearly wanted to downplay this finding so they wrote things like “our analyses show greater male variability, although the discrepancy in variances is not large.” Which is true in some sense but the point is that small differences in variance can make for big differences in outcome at the top. The authors acknowledge this with the following:
If a particular specialty required mathematical skills at the 99th percentile, and the gender ratio is 2.0, we would expect 67% men in the occupation and 33% women. Yet today, for example, Ph.D. programs in engineering average only about 15% women.
So even by the authors’ calculations you would expect twice as many men as women in engineering PhD programs due to math-ability differences alone (compare with the media reports above). But what the author’s don’t tell you is that the gender ratio will get larger the higher the percentile. Larry Summers in his infamous talk, was explicit about this point:
…if one is talking about physicists at a top twenty-five research university, one is not talking about people who are two standard deviations above the mean…But it’s talking about people who are three and a half, four standard deviations above the mean in the one in 5,000, one in 10,000 class. Even small differences in the standard deviation will translate into very large differences in the available pool substantially out.
If you do the same type of calculation as the authors but now look at the expected gender ratio at 4 standard deviations from the mean you find a ratio of more than 3:1, i.e. just over 75 men for every 25 women should be expected at say a top-25 math or physics department on the basis of math ability alone (see the extension for details on my calculation). Now does this explain everything that is going on? I doubt it. As Summers also pointed out it takes more than ability to become a professor at Harvard and if there are variance differences in characteristics other than ability (and there are) we can easily get a even larger expected gender ratio.
Does this mean that discrimination is not a problem? Certainly not but we need the media and academia to accurately present the data on ability if we are to understand how large a role other issues may play.
Addendum 1: Andrew Gelman points out that perhaps alone among the media, Keith Winstein at the WSJ reported the story correctly.
Addendum 2: The authors show variance ratios of 1.11 to 1.21, I take a VR of 1.16. If we set the female variance to 1 this implies the standard deviation for female ability is 1 and for male ability 1.077. Using an online calculator for the Normal distribution you can find that given their standard deviation .0102% of males have ability of 4 or greater (4 female sds) but given their sd only .0032% of females can be expected to have the same level of ability, thus a gender ratio of 3.18.
Note that we are assuming that mathematical ability is normally distributed – we know the data fit this distribution around the mean but we don’t know much about what happens at the very top.
More Inflation
First oil, then rice and wheat, and now the cost of hiring low-skilled labor is up by 12% in just one day. Shocking.
Hat tip to Don Boudreaux.
Bad News, Good News
Yesterday, I was supposed to be on Street Signs with Erin Burnett to talk about the effect of the Iraq war on the state of the economy. Sadly, they canceled me at the last minute. Bummer. Then when I got home, there it was, waiting, mocking, the Tivo recording of "my show." I felt bad but… I had to watch.
Heh, they got Joe Stiglitz to replace me! Well, anything less than a Nobel prize and I would have been insulted but I feel much better now! Amusingly, Stiglitz and I are good substitutes on this issue. In case you are wondering, we both think that at the present time the net effect of the Iraq war is (modestly) contractionary rather than stimulative due to higher oil prices, higher interest rates and less wealth.
Democrats Proudly Cut Medicare Benefits
Last week Congress cut benefits to Medicare recipients and liberal pundits applauded. Indeed, Paul Krugman said this was "Kennedy’s Big Day" and "the first major health care victory that Democrats have won in a long time." Of course, Krugman and the others who applauded this "victory" didn’t say that they were cutting Medicare benefits – even though that is exactly what they were doing – instead they framed the victory as one over privatization and waste. Here’s the story.
Medicare beneficiaries can enroll in Medicare’s fee for service plan or they can choose Medicare Advantage joining, for example, an HMO. In the latter case, Medicare pays the HMO a rate per enrollee and the HMO competes to obtain enrollees by offering them a package of benefits and premiums.
Now what you will be told about Medicare Advantage is that it is more expensive than traditional Medicare. Thus the CommonWealth Fund says:
Private Medicare Advantage (MA) plans were paid an
average 12.4% more per enrollee in 2005 compared with what the same
enrollees would have cost in the traditional Medicare fee-for-service
program…
That much is true. But why are MA programs more expensive? The answer, which one gets by innuendo and implication, is that Medicare Advantage programs are wasteful and the extra money is being pocketed by corporations.
The CommonWealth Fund says:
"…eliminating extra payments to private plans could save Medicare a projected $30 billion over five years." (italics added)
Paul Krugman says:
the fastest-growing type of Medicare Advantage plan, private
fee-for-service, costs taxpayers 17 percent more per beneficiary than
Medicare without the middleman. (italics added).
Robert Waldmann is least careful and in a comment on Tyler’s article on means testing says
Cowen doubts that expanding the public share of health insurance would
reduce costs. We have a test case medicare vs medicare advantage
accounts. They cost, on average 12% more per patient…
Thus the message is that traditional Medicare is cheaper because it eliminates the middleman, doesn’t involve private corporations, and is more efficient at lowering costs. None of this is true.
I’ll give you the full story in a minute but let me first point to one clue that something is amiss. According to all of the above "enrollment in these plans has been growing rapidly" (Krugman). Now why would so many Medicare beneficiaries opt out of low-cost, efficient Medicare and into high-cost, inefficient MA plans?
While you puzzle over the clue let’s cover the necessary background. Here is how the MA program pays a private provider (quoting the CBO).
Private plans that want to participate in the Medicare Advantage program must submit bids indicating the per capita payment for which they are willing to provide Medicare’s Part A (Hospital Insurance) and Part B (Supplementary Medical Insurance) benefits–and to take on the financial risk of doing so.
The government compares those bids with county level benchmarks that are determined in advance through statutory rules. The benchmarks are the maximum payment the government will make for enrollees in private plans; in most cases the plans’ bids (and the resulting payments) are lower than the benchmarks….
If a plan’s bid is less than the benchmark, Medicare pays the plan its bid plus 75 percent of the amount by which the benchmark exceeds the bid.
So far you might think that Krugman et al. have a point. If the benchmarks are set too high and Medicare pays the plan its bid plus 75% of the amount by which the benchmark exceeds the bid then the plans could bid their costs and get extra payments. Now, I hope that many of you are thinking, What about competition? Good thinking! Indeed, if that was all there was to it, competition would push the bids below costs. But in fact to resolve our puzzle we need not rely on competition and economic theory because here is the kicker (quoting the CBO again, italics added):
If a plan’s bid is less than the benchmark, Medicare pays the plan its bid plus 75 percent of the amount by which the benchmark exceeds the bid. Such a plan must return that 75 percent to beneficiaries as additional benefits or as a rebate of their Part B or Part D premiums.
Now the solution to our puzzle becomes clear. Why do beneficiaries choose MA plans?
…because such plans provide additional benefits beyond those available within traditional Medicare, including coverage for services not covered by FFS Medicare (for instance, dental services) and cash rebates of premiums or reduced cost-sharing.
In fact, the CBO estimates that the vast bulk of the increased payments to private providers flow to enrollees who get better benefits and lower payments. Indeed, in the case of HMOs enrollees benefit twice – first because the benchmarks are higher and second because, contra Krugman et al., the HMOs actually have lower costs than traditional Medicare! Thus the CBO writes:
In contrast, payments to HMOs averaged 10 percent above FFS costs…On average, HMOs offered extra benefits and rebates equal to 13 percent of FFS costs; those additional benefits and rebates reflected the difference between the benchmark (which averaged 10 percent above FFS costs) and the plans’ bids (which averaged 3 percent below FFS costs).
That could be written more clearly but what they are saying is that Medicare pays HMOs 10 percent more than they would pay for an enrollee in traditional Medicare but the HMOs offer the enrollee 13 percent more worth of extra benefits and rebates. In other words, the HMOs pass on to the enrollee all of Medicare’s "extra payments" plus some. (Note that this is exactly what one would expect in a competitive market.)
Now, I am not saying that higher Medicare payments are a good idea. But I dislike the fact that politicians are being lauded for fighting "wasteful privatization" when what they are really doing is cutting medical benefits for the elderly.
NFL Player IQ by Position Played
In Defense of Short-Selling
Props to Dean Baker.
Short-selling can play a very important role in the market. If
informed investors recognize that a stock is over-valued they perform a
valuable service by selling it short and pushing down its stock price.
This can both deprive the company of capital and be a signal to other
actors in the market that the company might not be as healthy as is
generally believed.The economy would have benefited enormously if large numbers of
traders had shorted Fannie and Freddie 4 years ago when they were
buying up hundreds of billions of mortgages issued to buyers who bought
homes at bubble-inflated prices. This would have stopped the bubble
years ago. Similarly, we could have prevented the financial chaos at
Merrill Lynch, Citigroup, Bear Stearns and the rest, if traders had
recognized their financial shenanigans and aggressively shorted their
stock. In the same vein, heavy shorting by informed investors could
have prevented the boom and bust of the tech bubble.The decision to intervene against short-selling is completely
inconsistent with the belief in the wisdom of the markets. Of course
short-sellers can be wrong and depress stock prices more than is
justified by fundamentals, but so what? The government doesn’t
intervene when it thinks investors have exaggerated the true value of a
stock. The public has no more reason to fear under-valued stock prices
than over-valued stock prices. This one-sided intervention by SEC is
hard to justify on any grounds.
Risk Free No Longer
Wow, we usually think about U.S. bonds as being the "risk-free" asset but with a credit default swap you can buy insurance on a US default and the price of such insurance is way up.
The change in price is a shock but to put things in perspective do note that the price for insuring US debt is now higher than for German debt but similar to that for Japanese and British debt. We are still far from Argentinian levels.
Hat tip to at, in the comments to my post on the peso problem.
Spend More Today
In Nudge Thaler and Sunstein motivate their Save More Tomorrow plan with the following unfortunate illustration:
Consider, for example, the case of Tony Snow, the former White House press secretary, who resigned at age fifty-two in 2007 to return to the private sector. He said his motivation for leaving was financial. "I ran out of money," he told reporters…Before serving as press secretary, Snow worked a much more lucrative gig as a Fox News Channel anchor. But he arrived at the White House not having learned Retirement 101 lessons. "Snow conceded: ‘As a matter of fact, I was even too dopey to get in on a 401(k).’
Sadly, Snow’s choices now look optimal. Ok, I know that may be in poor taste but let’s try to rescue this observation with some theorizing. Are we more likely to commit the error of saving too much or too little?
There are people who don’t save much because they have very low incomes, their behavior does not seem to be in error, especially when we take into consideration the various welfare programs that will cover people in their old age.
So let’s focus on people with moderate to high incomes. Thaler and Sunstein say that we are more likely to make errors when the benefits are upfront and the costs are delayed. Eating too much chocolate being a classic example. Ok, that suggests we may save too little.
T. and S. also argue that the less frequent a decision the more likely are errors. Frequency, however, cuts both ways – we only die once – so that’s a wash.
Over confidence and in particular the idea that we are special and will live a long life suggests the error is saving too much. Note that we also tend to think that our partner will be alive as well. My wife once asked me whether we were saving enough for "our" retirement. "Sure," I said, "don’t forget one of us will probably die before the other and I’m not saving for your future husband." "Why," she replied with a sigh, "can’t economists be more human?"
Availability bias probably also suggests we save too much – we see people who saved too little in the street but the ones who saved too much are dead and gone.
In theory, optimal saving equalizes the marginal utility of income across one’s lifetime – some programs like Kotlikoff’s ESPlanner attempt to calculate such an eqi-marginal utility flow and Kotlikoff’s finding is that a large fraction of Americans, some 40%, are saving too much. Kotlikoff’s program takes into account that we may need less wealth when we are old and retired (e.g. less transportation for work related reasons) but not that the marginal utility of wealth may be lower when we are old. (e.g. Money’s not so valuable if you don’t need it to or can’t use it to attract a mate.) Thus over-saving may be even more common than Kotlikoff suggests.
I do not know which error is more prevalent but if we are to be neither spendthrift nor miser we need to recognize both types of error.
Fannie Mae, Freddie Mac and the Peso Problem
A government bailout of the GSEs should not be a surprise. After all, for a long time the markets have been predicting that sooner or later there will be a very expensive bailout. What do I mean? According to Freddie Mac (quoting the OMB) "mortgage rates are 25 – 50 basis points lower because Fannie Mae and Freddie Mac exist in the form and size they do." Now, that is almost certainly an exaggeration but to the extent that interest rates are lower due to the GSEs some significant part of that is due to the market valuing the government’s implicit guarantee. In other words, interest rates are lower because the market is valuing the implied insurance. Now, the whole point of insurance is that sometimes the insurer must pay. Thus the market has been telling us all along that sooner or later the taxpayer was going to pay.
Maybe the taxpayers will have to pay today or maybe in some future tomorrow but the benefits of the GSEs are intimately tied to the costs – there is no such thing as a free lunch. The lunch may look free for a long time – as in the classic peso problem – but what that means is that when the bill comes due it will be big.
Taliban v. Coase
Sadly, the Taliban are succeeding where Coase (and the Pakistan government) have failed.
The mountain of white marble shines with such brilliance in the sun
it looks like snow. For four years, the quarry beneath it lay dormant,
its riches captive to tribal squabbles and government ineptitude in
this corner of Pakistan’s tribal areas.But in April, the Taliban
appeared and imposed a firm hand. They settled the feud between the
tribes, demanded a fat fee up front and a tax on every truck that
ferried the treasure from the quarry. Since then, Mir Zaman, a
contractor from the Masaud subtribe, which was picked by the Taliban to
run the quarry, has watched contentedly as his trucks roll out of the
quarry with colossal boulders bound for refining in nearby towns.
Mr Fantastic
China Fact of the Day
China is the world’s largest importer of chicken feet and the United States is the world’s largest exporter. Tyson Foods alone send some 2.8 billion chicken feet to China every year. The chicken feet are sold at Chinese Wal-Marts (among other places) which in China are upscale and appreciated for their high quality American goods.
All Tyler, All the Time
The tyler-city blog is a computer generated blog of gibberish meant solely to generate links (can readers explain the economics?). It is drawn from all over, including quite a bit from Marginal Revolution. It’s gibberish but as Tyler might say even a million monkeys occasionally generate some very good sentences. Here are a few, I’ve provide links but please don’t encourage them too much.
I thought both were tyler cogent, for quite complex topics.
The Ton Ball That Keeps The Taipei tyler Tolerant Is Pretty
and it is good to know that "besides his many talents, Tyler was also a really nice kid." But my favorite posting is this one:
A Girl Named Florida
I’ve been reading Leonard Mlodinow’s The Drunkard’s Walk: How Randomness Rules our Lives. The book covers the Monty Hall problem, Bayes’s Theorem, availability bias, the illusion of control and so forth. If these are unfamiliar, look no further for an entertaining account.
On the other hand, I can’t say that I learned much I didn’t already know. Nevertheless, I still enjoyed reading the book – it’s well written and filled with interesting nuggets (Did you know that the great mathematician Paul Erdos refused to believe that you should switch doors?). If you teach probability theory or intro stats you will find lots of good examples to brighten up your lectures.
One problem did intrigue me. Suppose that a family has two children. What is the probability that both are girls? Ok, easy. Probability of a girl is one half, probabilities are independent thus probability of two girls is 1/2*1/2=1/4.
Now what is the probability of having two girls if at least one of the children is a girl? A little bit harder. Temptation is to say that if one is a girl the probability of the other being a girl is 1/2 so the answer is 1/2. That’s wrong because you are not told which of the two children is a girl and that makes a difference. Better approach is to note that without any additional information there are four possibilities of equal likelihood for the sex of two children (B,B), (G,B), (B,G), (G,G). If we know that at least one is a girl we can remove (B,B) so three equally likely possibilities, (G,B), (B,G), (G,G), remain and of these 1 has two girls so the answer is 1/3.
Ok, now here is the stumper. What is the probability of a family having two girls if one of the children is a girl named Florida?
At first it seems impossible that knowing the name should make a difference. Surely, the answer is 1/3 just as before? After all, every child has a name. But knowing the name does make a difference. Here’s a hint, Florida is a rare name.
Onion Futures
There are none.
The bulbous root is the only commodity for which futures trading is
banned. Back in 1958, onion growers convinced themselves that futures
traders (and not the new farms sprouting up in Wisconsin) were
responsible for falling onion prices, so they lobbied an up-and-coming
Michigan Congressman named Gerald Ford to push through a law banning
all futures trading in onions. The law still stands.And yet
even with no traders to blame, the volatility in onion prices makes the
swings in oil and corn look tame, reinforcing academics’ belief that
futures trading diminishes extreme price swings.
Amazing, onion farmers and Congress panic in 1958 with the Senate Committee arguing that
…speculative activity in the futures markets causes such severe and unwarranted fluctuations in the price of cash onions…[that a] complete prohibition of onion futures trading [is necessary] in order to assure the orderly flow of onions in interstate commerce…
and for going on fifty years onion futures are banned. Makes me want to cry.
More here on the banning of futures markets . (A report from 1956 indicates that the fluctuations at that time were due to an attempted swindle.)
Hat tip to Newmark’s Door