Sometimes rational behavior means nobody has any idea why they are doing something

The always-excellent Planet Money podcast recently did an episode on why milk is always at the back of the grocery store. It’s a fantastic piece, and well worth the 16-minute listen, but can be summarized pretty briefly. It turns out that there are two theories for the milk-in-the-back phenomenon: exploiting behavioral economics and cost minimization. The behavioral economics story is the one I was more familiar with: milk is an extremely common purchase, and it is placed in the back in order to force people to walk past a range of tempting items. Since consumers are prone to impulse purchases, this induces them to spend more.

The cost-minimization theory is one I wasn’t familiar with, but it rings true. Milk is extremely prone to spoilage and must be kept consistently cold to keep it drinkable. Keeping it in the back is the cheapest way to maintain the “cold chain” needed for this; moving it to the front would cause more losses due to spoilage and/or require.

That’s the debate, but the best part of the episode is that no one knows what the right answer is. They talk to a range of experts who voice various opinions on the subject, and get support for both theories. The first interview is with a guy who is the dairy buyer for a major grocery store chain. He voices support for the tempting-consumers theory, and literally says he “believes” that that’s why they do it. He is the guy that is doing this! And he believes that the reason why he puts the milk in the back is that it tempts the customer to spend more. The cold chain theory doesn’t fare a ton better either, getting barely more than half of the votes from people who are in the business of selling milk.

None of this implies that the people who run grocery stores are not behaving optimally, however. You don’t need to understand your own strategy to maximize profits, or even your own well-being. Imagine you run a big grocery store, or even just the dairy section of one: you make tons of decisions and face hundreds of constraints. And you observe what happens to costs, volumes sold, and profits, whenever you change something. So your process has ended up a certain way, and you can legitimately not know the exact reason why. You know that if you do different stuff with the milk, profits go down – from your own experience, from watching other stores, etc. You keep doing what you’re doing, because you are sitting at an optimum, but you don’t actually know why.*

My gut instinct is that this kind of rational behavior – where people are at an optimum but don’t know exactly why – is exceedingly common. I am reminded of the complex traditions around using one’s signals on unlit roads in rural Africa. If you approach traffic headed in the opposite direction, you have to lower your high-beams and turn on your right turn signal (the one closest the the approaching car). I’ve heard various of explanations for this: that it reminds people to dim their headlights, that it informs approaching cars about the location of the edge of your vehicle, etc. No one knows why they do it, they just do. Realizing that there has to be some good reason (or reasons!), when I drive on dark roads in Malawi, I do the same thing.

*And of course there might be multiple reasons – a reasonable model of dairy section location would involve the firm minimizing the costs of all its items and balancing that against the value of all its sales.
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How to reduce the incidence of rape using public policy

Rape is a big problem in the United States. Reliable figures are very hard to get for such a stigmatized and personal crime, but regardless of the exact prevalence, it happens much too often for us to be comfortable with. Well-publicized cases out of India suggest that in developing countries the problems are even worse (but reporting is probably worse as well).

Although most people agree that rape is an important issue, there is a dearth of actual practical policy suggestions for dealing with it. Encouraging women to be more careful and aware of their surroundings smacks of victim-blaming and is unlikely to do much to reduce all-too-common acquaintance rapes and date rapes. Pointing the finger at “rape culture” could be part of the right answer, but it’s not obvious what to do about that culture, and the people who make major contributions to rape culture probably aren’t reading think pieces about it. More important, neither of these proposed solutions is backed by actual evidence.

However, an excellent new paper by Scott Cunningham and Manisha Shah does provide evidence about something that works to decrease rape: decriminalized prostitution, specifically “indoor” prostitution (massage parlors, escort services, and online sex work). Rhode Island unintentionally decriminalized the practice in 2003 (they have an excellent discussion of how this happened), and Cunningham and Shah show that this increased the size of the sex market and also reduced cases of gonorrhea (which is mainly spread by sex workers) by 39% across the whole population. Even more striking, they demonstrate that the overall incidence of rape declined by 31%.

One of the best features of the paper is that they use what I once heard Steve Levitt refer to as a “mosaic of evidence” to support their claim. They draw on a range of different datasets to illustrate their findings both visually and numerically, and to buttress their assumptions. Their main estimates are based on a difference-in-differences between Rhode Island and the rest of the US, but these are buttressed by a “synthetic control” approach. The synthetic control method, from Abadie and Gardeazabal (2003), finds other states that look similar in terms of previous trends for the outcome, and uses a weighted average of them to form a comparison group. The authors provide tons of graphs to show exactly what they are doing, and they all look about as compelling as this one does:

Rapes over time

What’s especially striking about this graph is that Rhode Island re-criminalized prostitution in 2009, exactly when rape cases go back up.

Cunningham and Shah are very careful to say that they cannot conclude exactly why decriminalizing prostitution reduces cases of rape. They consider a number of potential mechanisms, and conclude that the most likely one is that, for some men, rape and prostitution are substitutes. That is, men commit rape in part due to sexual desire, which can be satisfied in other ways. While Cunningham and Shah’s paper cannot demonstrate this for sure, their finding is consistent with other research by Todd Kendall that finds that the rollout of the internet, and the attendant increase in the accessibility or pornography, appears to have driven a decrease in cases of rape.

One potential issue with decriminalized prostitution is that it may just be substituting one form of oppression of women for another: decriminalized indoor prostitution may attract abusive pimps or madams, or encourage the illegal sex trade. The experiences of Nevada and several countries in Europe suggest a solution, however – full legalization and regulation of sex work would allow for law enforcement to protect women who engage in sex work, and undermine the market for illicit paid sex. A similar issue potentially affects pornography, where some actresses are exploited (but some emphatically state that they are doing what they love). Here, too, the answer is to have a legal but well-regulated market: exploitation may happen where pornography is legal, but if it were forced underground and run by criminal organizations, it would probably be much worse.

Cunningham and Shah’s paper, along with the earlier Kendall paper, suggest two ways to reduce rapes through public policy: legalized access to pornography, and legalizing sex work. These policies would both operate through the same mechanism – which is that some men commit rape due to sexual urges. That’s not an idea that we are particularly comfortable with as a society – a common claim is that rape is an act motivated by power, or anger, and not by sexual gratification. But if we want to reduce the incidence of this awful crime, we need to move out of our comfort zone, and rely on the evidence rather than what we’d like to believe.

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Odds ratios are a catastrophe

Adam Larson sent me the following question about a study of obesity and a press release about it from NPR. The claim, made in both the press release and the underlying article, is that weight discrimination makes the already obese 3 times as likely to remain obese, and the non-obese 2.5 times as likely to become obese. Adam writes:

They interpret odds ratios of 2.5 and 3 as “2.5 times as likely” and “3 times as likely”.

Balderdash, yes? I assume what they’re getting at is that in one group something like 85% remained obese; in the other 75%. This gives an odds ratio of (.85/.15)/(.75/.25)=1.89

So common sense would call it a 10 percentage point decrease or a 12% decrease, right?

Adam is spot-on. An odds ratio is the odds of an event happening for one group divided by the odds of a thing happening in another. Odds are summaries of probabilities that get used by sports books and nearly no one else, because they are counter-intuitive non-linear approximations to probabilities. If an event has an X% chance of happening, the odds that it happens are (X%)/(100-X%). The basic problem with odds ratios is that long ago someone (we should figure out who and curse their name) realized that for rare outcomes, an OR is approximately a relative risk, or (% chance thing occurs in treatment group)/(% chance thing occurs in control group). That is:

(0.01/0.99)/(0.02/0.98) ≈ 0.5 = 0.01/0.02

That has ever since been taught to applied statisticians working in certain fields (public health is one example) who use odds ratios for the scientifically important reason that they are the default output of many regression packages when you run a logistic regression.*

And so people misinterpret them constantly, presently odds ratios as relative risks even when they are not small, and the approximation does not hold. This is even before we get into the fact that calling a change from P=0.01 to P=0.02 a “100% increase in risk” is itself fairly absurd and misleading. It’s a one percentage-point increase. There is no intrinsic sense in which “the risk tripled” actually means anything. Did you know that if you go in the ocean you are infinity times more likely to get eaten by a shark than if you stay on land? (You probably did, but it’s a stupid number to think about. What is actually relevant is that the absolute risk went up by some fraction of a percentage point.)

For this paper, under the assumption that their regression adjustment doesn’t change too much, we can actually back out what the percentages really are. First, the effect on the not-initially-obese:

Mean outcome = (% discriminated)*(mean for discriminated people) + (1 – % discriminated)*(mean for non-discriminated people)
0.058 = 0.08X + 0.92Y
Odds ratio = ((mean for discriminated people)/(1 – mean for discriminated people)) / ((mean for non-discriminated people)/(1 – mean for non-discriminated people))
(X/(1-X))/(Y/(1-Y)) = 2.54
Y = (50X)/(127-77X)
0.058 = 0.08*X + 0.92*(50X)/(127-77X)
X = 0.1230
Y= 0.0523

So the change is 7.2 percentage points. Put less clearly, P(became obese) has gone up by a factor of 2.35 for those who experienced weight discrimination, relative to those who did not. That is different from the OR of 2.54, but their figure isn’t too far from the relative risk.

Repeating the process for their other analysis, however, reveals how misleading ORs can be:
0.263= 0.08X + 0.92Y
(X/(1-X))/(Y/(1-Y)) = 3.20
Solving these equations for X and Y gives us:
X = 0.505
Y= 0.242

Here the risk ratio is 2.08, not 3.20. The percentage-point change of 26.3 remains completely comprehensible, as it always is. Misusing odds ratios here allowed them to overstate the size of their effect a factor of 50%! I suspect, but am not sure how to prove, that with regression adjustment these figures could look even more misleading.

As most people who read this already know, even if presented correctly the figures wouldn’t mean anything. There’s no reason to believe the relationship being studied is causal in nature. Indeed, it probably suffers from classic reverse causality: people who are gaining weight (or failing to lose weight) are likely to perceive a greater degree of weight discrimination. But presentation matters too. First, clear presentation can help us make use of studies, even when they are as limited as this one is. As the above derivation illustrates, figuring out what an odds ratio actually means involves 1) the annoying process of scrounging through the paper for all the variables you need and 2) solving a system of two equations for two unknowns, which most people can’t do in their head. This detracts very substantially from a paper’s clarity: in general, when I see odds ratios presented in a paper, I have no idea what they mean. An OR of 2 could mean that the risk went from 1% to 2% or (to use a variation on Adam’s example) from 75% to 86%, or a whole host of other things.

Second, poor presentation has consequences. Health risks are often reported using relative risks, or, worse yet, using ORs that are presented as relative risks. This is often extremely misleading, since a doubling of risk could mean that the chance went from 0.001% to 0.002%, or from 50% to 100%. Misleading and confusing people about risks undermines the basic goal of presenting health risks in the first place: to help people make better decisions about their health.

*I honestly believe that if we made mean marginal effects the default, and forced people to do ORs and AORs manually, they would disappear within 10 years. Being forced to construct ORs manually would also force people to understand what they are, which would stop people from using them.
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What empirical microeconomics tells us about reparations

Ta-Nehisi Coates argues that the United States government should pay reparations to African-Americans for slavery and institutionalized racism. The essay is long and full of supporting evidence, and generally makes a strong case that the US government bears responsibility for oppressing blacks for hundreds of years. While Coates digresses occasionally  – into claims of broader guilt by all Americans, or all whites, or into arguments that America’s current prosperity depends on its history of oppressing blacks – those claims are not necessary for his main point to hold water. That point is fairly straightforward: the US government was complicit in a moral evil, and it should take steps to make right for that evil, as it did, for example, for the internment of Japanese Americans during World War II.

Leaving aside the merits of the underlying idea, and the tasking of pinning down what the value of the reparations would be and how to allocate them, I wanted to discuss the practical aspect: what would providing reparations accomplish? Could transferring money to blacks help close the yawning gaps between them and whites that exist across a broad range of social indicators? Reparations need not be cash transfers – Coates cites Charles Ogletree’s idea of reparations in the form of job training programs – but usually the term is associated with the payment of cash to the afflicted group. This fixes a key economic question: what would happen if the US government made a massive financial transfer to every black person in America?

In some sense the right answer to this is “we don’t know”. We have never tried doing this, let alone in an experimental framework that would allow us to measure its effects. Coates does list one empirical example – the German payment of Holocaust reparations to the Israeli government, which is credited with funding the country through a tough spell and contributing to substantial economic growth. But most of those payments went to the government, not to individuals, so it is unclear how those effects would translate to the context of reparations to blacks in the US.

Even though no one has ever run this experiment, we do have evidence on what happens when people receive large cash transfers. The best evidence comes from a paper by Hoyt Bleakley (who is joining Michigan’s economics faculty in the fall) and Joseph Ferrie, about a lottery that distributed land at random to adult white males in Georgia.* The winners of this lottery received land worth approximately as much as the entire wealth holdings of the median person at the time. Given that the average black family has one sixth the wealth of the average white family, this is actually pretty close to the magnitude of the transfer we’d be talking about.


This image (from Wonkblog) shows that the black-white wealth divide has widened rather than narrowing over time

Large cash transfers help: they make the recipients richer. But they don’t have the long-term social ramifications that you might hope for. The children and grandchildren of lottery winners end up no wealthier and no better-educated than non-winners. The big caveat with this comparison is that the Bleakley and Ferrie paper studies people from the 19th century, so the sample and context are quite different than they are today. However, I’d actually expect those differences to lead to larger effects than we’d see from targeting a poorer, more disadvantaged group. Overall, this suggests that wealth transfers – even massive ones – will not have transformational effects on socioeconomic status that last across generations.

On the other hand, a wide range of evidence suggests that, contrary to stereotypes, people (even poor people) do not “waste” cash transfers on alcohol, cigarettes, or other vices.** Those results are for transfers on a scale much smaller than reparations would operate on, and are for much poorer populations than the typical black American. But implicit in the the stereotype that money will go toward alcohol is notion that poorer people should have bigger problems with this.*** Since even very poor people seem to have no problem refraining from potentially-problematic spending, it is unlikely that this would be an issue for a reparations program.

Taken together, the evidence from empirical economics tells us that reparations, if done as pure financial transfers, would make blacks richer and with few downsides – but that they would not have transformative effects on the long-run gaps in outcomes between whites and blacks. While wealth is inherited, wealthy people also propagate success through their family lines by passing down other attributes – from education to behaviors to social connections to their race – that end up washing out the effects of wealth alone. To fix the black-white gap in a permanent way, we need to address all sorts of other differences as well; addressing wealth alone is not enough.

What about other ways of providing reparations? The literature on job training programs for marginalized groups is fairly discouraging, so I’m not convinced that Ogletree’s proposal would work well (although maybe we need to work on developing better job training). Another possibility is to work through the education system. Roland Fryer’s research has shown that improving middle-school educational outcomes for blacks helps them close gaps in other social outcomes. At the college level, there is robust, although not necessarily causal, evidence that high-quality colleges help blacks quite a bit (and matter much less for whites). One policy that might work is to replace affirmative action with an official reparations program, funded by the federal government, that creates additional slots at all universities to accommodate black students. This would reduce the racial tension that is stirred up by the current system, where people perceive that they are being denied admission based on their race, and where the moral and legal justification for the scheme is not made clear. It might reduce opposition to the program as well.

More broadly, we still need more evidence about what kinds of programs help generate permanent reductions in the black-white social divide. If reparations end up being taken seriously, then the government should fund and promote experimental and regression-discontinuity research into a wide range of possible programs in order to see which ones work. Financial transfers alone may not work – but we have the empirical tools needed to figure out what does.

*In a dark irony, this land came from one of the worst crimes against humanity the US government has ever committed.
**As I’ve discussed on this blog in the past, it is not obvious that these purchases are wasteful; we need to take seriously the idea that people have agency – that they can be trusted to make their own decisions.
***Basic economic theory actually suggests the opposite, since poorer people have a tighter budget constraint. But it also tells us that people’s unaltered choices maximize their own welfare, so this is a non-problem. Chris Blattman believes that the homeless in the US are fundamentally different from similar-looking populations in Africa, but that would only apply to the poorest black people who received reparations.
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Data that confirms my priors: infrastructure edition

In our panel of 38,427 subnational regions from 126 countries with yearly observations from 1992 to 2009, we find that subnational regions have more intense nighttime light when being the birth region of the current political leader.

From “Regional Favoritism”, by Hodler and Raschky in the Quarterly Journal of Economics.

One of my favorite anecdotes about infrastructure in the developing world is that when I was collecting data near the Jali trading center in Southern Malawi, there was no running water but the cell phone-based internet was competitive in quality with what Comcast provided back in Ann Arbor. Jali had its own cell tower for MTN, a rarely-used, government-sponsored cell network that provides the best (albeit priciest) internet in the country. I later learned that Thyolo, another relatively out-of-the-way town, also has such a tower. Mulanje, a much bigger boma near the border with Mozambique, doesn’t. Why? Well, one possible explanation is that Thyolo is the home of the late president, Bingu wa Mutharika, and his wife has a house in Jali.

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Is the top 1% of the income distribution 98% male?

Matt Yglesias says yes, noting that “the overwhelming maleness of the top 1 percent is also interesting in a world where some people are proclaiming that the economy is working great for women.” He immediately backs off a bit, noting that of course most of the men in the top 1% are married (over 90% of them, in fact) – so a lot of the benefits of being a one percenter accrue to women as well. But is that what the data actually tell us? Here is the table in question, from an article by Lisa A. Keister in the Annual Review of Sociology.


If this table says that the top 1% are 98% male, it also says that the bottom 90% are 70% male. We could also conclude that the US as a whole is 72.9% male. Of course that’s false – the Survey of Consumer Finances is a household-level survey, not an individual-level one. Presumably, then, these data are all for household heads (but they could be for whoever happened to answer the survey – I couldn’t find that in the original article).

The author of the article doesn’t handle this much better either, saying that “this table shows that members of the one percent are disproportionately male, white, and married”. It does not. What it does show – probably – is that households in the one percent are disproportionately male-headed. And it also strongly suggests that men lie further up the income scale than women. But we can’t look at this table and conclude that 98% of people in the top 1% of the income distribution are male.

EDIT: Yglesias’s updated post removes the assertion that the 1% are 98% male, but I’m leaving this up for posterity (and because the underlying article still makes a similar claim).

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Bogus Complaints about the use of Discrete Variables

Orazio Attanasio and Valérie Lechene (A&L) have an excellent article in the latest Journal of Political Economy that exploits the randomized rollout of PROGRESA to test the collective model of household consumption. Their analysis rests of the fact that once we condition on total consumption, the only way PROGRESA should plausibly affect the shares of consumption allocated to specific goods is through increasing women’s bargaining power (PROGRESA transferred money directly to children’s mothers). They make a similar argument for another variable, the relative strength of the two spouses’ family networks – these two variables, which affect consumption shares of different goods only through bargaining power, are called “distribution factors”. The collective model of household consumption states that however resources are allocated within the household, there is no waste; this is equivalent to saying that there is a unique index, called the Pareto weight or the sharing rule, that governs how all the distribution factors affect the shares of spending allocated to different goods. All distribution factors, then must enter the demand system proportionally, so we can effectively condition on one of them and explain away all the others. If demand shares still depend on a second distribution factor after we appropriately condition on the first, we can reject the collective model. They find that the collective model does not fail this test, while the simplistic unitary model is easily rejected (since PROGRESA changes consumption patterns, conditioning on total expenditure).

A&L exemplifies what we want to accomplish through conducting field experiments in economics: they combine a deep understanding of the institutional and cultural context of the experiment with an equally thorough analysis of what various economic models tell us about what should happen. As a result, A&L aren’t just estimating a parameter consistently or measuring the impact of a program, they are advancing our knowledge of how consumers make decisions – and in an empirically credible fashion. It’s also extremely well-written; I can hardly do it justice via a brief summary.

The only arguable shortcoming of the paper is that they make much of the fact that one of the distribution factors they rely on, the relative strength of the two spouse’s family networks, on is continuous. Continuity of at least one distribution factor is a formal requirement for the mathematics of their argument to go through. The problem with this claim is that it is false. The number of family members has only a finite number of points of support, thus leading to a finite number of potential values for the variable. The same even applies to their alternative measure, which uses the total consumption of each spouse’s family network. Money is “more” continuous than counts of people, sure – but it is not actually continuous.  This doesn’t really undermine their argument, which is that you can’t use the PROGRESA treatment if a continuous variable is needed. PROGRESA treatment is, by definition, binary, and hence discrete. It definitely seems more valid to use something that is arguably a discretized proxy for an underlying continuous variable: although we observe only discrete ratios of numbers of people, that in principle could be measuring a variable that is actually continuous.

Unfortunately, stating that their alternative variable seems more valid is about as far as I think we can go. I’m not aware of any proof that having a “mostly continuous” variable is “good enough”, nor even that having things be “more continuous” is “better”.* This is a very general problem: most of economic theory, and most of the math underlying econometrics, technically requires the variables we are working with to be continuous. But all of the variables actually used in empirical economics are discrete: the minimum granularity of money is cents (or arguably mills); for time, we never measure anything below seconds.**

None of this means that the mathematical and statistical tools we use don’t work. On the contrary, they seem to work just fine even when things are obviously discrete. The canonical example of ignoring discreteness is the “linear probability model”, which has been rehabilitated in the eyes of economists (in particular Josh Angrist). We seem to have learned, as a discipline, that if the marginal effects computed by a probit are meaningfully different from those that come out of an LPM, the solution is to fix your specification rather than to trust that the error term is normally-distributed. I’ve personally learned that pretending things are continuous is also fine in other contexts – for example by learning how to implement a count model on some of my data only to find that its estimates of marginal effects were identical to OLS to the 4th decimal place.

Pointing out discreteness as a statistical concern, or an issue with someone’s model, is usually just a cheap “gotcha”. Yes, it’s technically a problem. But it’s technically a problem with every economics paper that uses continuity – which is a lot of papers. As a discipline, economics seems to be strikingly inconsistent on whether we worry about continuity. We usually ignore it when working with discrete quantities like money or hours worked or years of education or test scores,  and nobody complains. There’s no good reason to criticize the use of variables that are discrete at the level of whole numbers while not objecting equally to the use of variables that are discrete at the level of hundredths of a whole number.

* That doesn’t mean that there is no such proof. However, if such a proof exists, Attanasio and Lechene don’t cite it, and neither do other researchers who insist that relatively more-discrete variables are more problematic.
** Broadly-informed readers might also note that according to the best of our knowledge, almost nothing is actually continuous, which doesn’t do much to limit our ability to use calculus to understand the physical world.
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