Alex Tabarrok

The New York Times covers a controversy about a Texas history textbook:

Coby Burren, 15, a freshman at a suburban high school south of here, was reading the textbook in his geography class last week when a map of the United States caught his attention. On Page 126, a caption in a section about immigration referred to Africans brought to American plantations between the 1500s and 1800s as “workers” rather than slaves.

textbook caption

The black lives matter movement is upset that slaves are referred to as workers.

I am upset that the caption is factually incorrect even if rewritten not to use the word worker. In particular, it is not true that millions of slaves were brought from Africa to the southern United States. In fact, less than half a million came to the United States.

Here is Henry Louis Gates  Jr:

The most comprehensive analysis of shipping records over the course of the slave trade is the Trans-Atlantic Slave Trade Database, edited by professors David Eltis and David Richardson. (While the editors are careful to say that all of their figures are estimates, I believe that they are the best estimates that we have, the proverbial “gold standard” in the field of the study of the slave trade.) Between 1525 and 1866, in the entire history of the slave trade to the New World, according to the Trans-Atlantic Slave Trade Database, 12.5 million Africans were shipped to the New World. 10.7 million survived the dreaded Middle Passage, disembarking in North America, the Caribbean and South America.

And how many of these 10.7 million Africans were shipped directly to North America? Only about 388,000. That’s right: a tiny percentage.

Here is the primary database for what we know about the Atlantic Slave Trade which lists 305,326 slaves brought to the USA. Gates goes on to note that some 60-70 thousand slaves initially brought to the Caribbean ended up in the United States so he estimates that perhaps 450,000 African slaves in total were brought to the U.S. over the course of the slave trade.

If you want to understand the slave trade it’s important to understand that the vast majority of the slaves taken from Africa were shipped to the Caribbean and South America. If you want to understand slavery in America it’s important to understand that most slaves in the United States were born into slavery. Also, as Gates notes, it’s a rather striking and amazing fact that “most of the 42 million members of the African-American community descend from this tiny group of less than half a million Africans.”

Regardless of whether you think that slaves are workers or not the textbook failed its students by getting the facts wrong. In a better culture, that failure would make for a controversy and a story in the New York Times.

Hat tip: Arthur Charpentier on twitter.

Ayn Rand and The Martian

by on October 5, 2015 at 7:25 am in Books, Film, Philosophy, Science | Permalink

The Martian is the most Randian movie in years, perhaps in decades. Ayn Rand is best known for her defense of capitalism but her defense of reason was even more fundamental to her thought. The Martian has no bearing on politics but it reminded me of Rand’s essay on Apollo 11 and the moon landing, the launch of which she witnessed Apollo 11 - 2from Kennedy Space Center.

Rand wrote that the Apollo 11 mission “conveyed the sense that we were watching a magnificent work of art – a play dramatizing a single theme: the efficacy of man’s mind.” The  efficacy of man’s mind and the power of reason is exactly the theme of The Martian.

As Rand continued:

That we had seen a demonstration of man at his best, no one could doubt…And no one could doubt that we had seen an achievement of man in his capacity as a rational being–an achievement of reason, of logic, of mathematics, of total dedication to the absolutism of reality.

The difference is that Apollo 11 gave the sense that we were watching a magnificent work of art but it was real. While the Martian gives the sense that we are watching something real but it is a magnificent work of art. Have we not been diminished? Nevertheless, the sense of life of the event and the movie are the same and the movie is gripping, thrilling and uplifting, a triumph for Ridley Scott and the author, Andy Weir.

Addendum: See Tyler’s review as well.

Corporate Prediction Markets Work Well

by on October 2, 2015 at 7:27 am in Economics | Permalink

Prediction markets predict public events such as election outcomes better than do polls or other forecasting mechanisms. Internal corporate prediction markets in events such as sales forecasts, product launch times, and product feature demand have been less well studied. Internal corporate markets tend to have fewer participants than public markets and the participants often have strategic interests and biases. Thus, it has been an open question how well these markets operate.

Cowgill and Zitzewitz report on a number of different types of prediction markets run by Google, Ford and Firm X and although they find evidence for some biases they also find that corporate prediction markets also work better than alternative forecasting methods.

Despite large differences in market design, operation, participation, and incentives, we find that prediction market prices at our three companies are well calibrated to probabilities and improve upon alternative forecasting methods. Ford employs experts to forecast weekly vehicle sales, and we show that contemporaneous prediction market forecasts outperform the expert forecast, achieving a 25% lower mean-squared error (p = 0.104).

…The strong relative predictive performance of the Google and Ford markets is achieved despite several pricing inefficiencies. Google’s markets exhibit an optimism bias. Both Google and Ford’s markets exhibit a bias away from a naive prior (1/N, where N is the number of bins, for Google and prior sales for Ford). However, we find that these inefficiencies disappear by the end of the sample. Improvement over time is driven by two mechanisms: first, more experienced traders trade against the identified inefficiencies and earn higher returns, suggesting that traders become better calibrated with experience. Secondly, traders (of a given experience level) with higher past returns earn higher future returns, trade against identified inefficiencies, and trade more in the future. These results together suggest that traders differ in their skill levels, they learn about their ability over time, and self-selection causes the average skill level in the market to rise over time.

Addendum: It’s an interesting commentary on academic publishing that Marginal Revolution first covered this paper in a working version in 2008! An extended version was received by the Review of Economic Studies in 2010 which accepted a final version in 2014 and then published the paper in 2015.

Transport for London is preparing to launch a crackdown on Uber, proposing a series of new rules that will hit the popular minicab-hailing app in one of its most popular cities.

…The proposals include a minimum five-minute wait time between ordering a private hire vehicle and it arriving, and banning operators from showing cars for hire within a smartphone app – a hallmark of the American company’s service.

No, this is not from an Ayn Rand novel.

These proposed rules so nakedly protect rent-seekers and make life worse for consumers that I don’t think they will succeed. Even if the rules fail, however, we shouldn’t be complacent about the dangers to innovation.

What made Uber different and controversial is that their Ayn Rand loving CEO followed the adage that it’s better to ask for forgiveness than permission. Uber skirted the law and went to consumers directly about whether they wanted transportation innovation. Consumers around the world responded with a resounding Yes to the Uber-referendum so regulators and rent-seekers who want to control Uber now must also fight Uber-consumers. That genie won’t go back into the bottle.

In the usual scenario, however, innovation can be quashed before consumers have a chance to know what they are missing. Had the taxi companies had an inkling of what was coming it would have been easy to to pass stricter laws in advance that would have made Uber impossible to get off the ground. Of course, in many industries today the old guard does have an inkling of what is coming and that should frighten anyone who wants to see greater innovation.

Asymmetric Information and Signaling

by on September 29, 2015 at 7:25 am in Economics, Education | Permalink

MRUniversity now has its own video production team! We are continuing to work with the artists at Tilapia films to produce outstanding videos for teaching economics–these videos work great with our textbook, Modern Principles. Our in-house production team will be working to polish our “regular” videos in a way that enhances the learning experience. You can see an example of the new style in the video below featuring Tyler on signaling.

Our Principles of Economics course is now complete. You can guess what is coming soon!

The drug Daraprim was increased in price from $13.60 to $750 creating social outrage. I’ve been busy but a few points are worth mentioning. The drug is a generic and not under patent so this isn’t a case of IP protectionism. The story as I read it is that Martin Shkreli, the controversial CEO of Turing pharmaceuticals, noticed that there was only one producer of Daraprim in the United States and, knowing that it’s costly to obtain even an abbreviated FDA approval to sell a generic drug, saw that he could greatly increase the price.

It’s easy to see that this issue is almost entirely about the difficulty of obtaining generic drug approval in the United States because there are many suppliers in India and prices are incredibly cheap. The prices in this list are in India rupees. 7 rupees is about 10 cents so the list is telling us that a single pill costs about 5 cents in India compared to $750 in the United States!

drugs India

It is true that there are real issues with the quality of Indian generics. But Pyrimethamine is also widely available in Europe. I’ve long argued for reciprocity, if a drug is approved in Europe it ought to be approved here. In this case, the logic is absurdly strong. The drug is already approved here! All that we would be doing is allowing import of any generic approved as such in Europe to be sold in the United States.

Note that this in not a case of reimportation of a patented pharmaceutical for which there are real questions about the effect on innovation.

Allowing importation of any generic approved for sale in Europe would also solve the issue of so-called closed distribution.

There is no reason why the United States cannot have as vigorous a market in generic pharmaceuticals as does India.

Hat tip: Gordon Hanson.

A new paper from Fernandez, Gohmann and Pinkston shows that counties in Kentucky that forbid alcohol have more meth labs than otherwise similar counties. I like the research but in truth any paper with both Breaking Bad and Justified references is a winner in my book.

Abstract: This paper examines the influence of local alcohol prohibition on the prevalence of methamphetamine labs. Using multiple sources of data for counties in Kentucky, we compare various measures of meth manufacturing in wet, moist, and dry counties. Our preferred estimates address the endogeneity of local alcohol policies by using as instrumental variables data on religious affiliations in the 1930s, when most local-option votes took place. Alcohol prohibition status is influenced by the percentage of the population that is Baptist, consistent with the “bootleggers and Baptists” model. Our results suggest that the number of meth lab seizures in Kentucky would decrease by 24.4 percent if all counties became wet.

The authors suggest that alcohol users who buy alcohol in places where it is banned become acculturated and familiar with illegal networks making it easier for them to buy meth. In a reverse of the usual story, alcohol prohibition becomes the gateway to other illegal activities.

In my interpretation, however, the association of meth labs and alcohol prohibition is due more to supply side factors than demand side factors. In particular, a long history of moonshine production in dry Kentucky counties leads to an accumulation of knowledge about where to hide the labs, how to evade the law and who to bribe. In this version of the theory, lifting alcohol prohibition doesn’t necessarily reduce meth production because the knowledge and the networks remain in place.

A modified version of the theory can combine demand and supply factors. If there are economies of scope between alcohol production and meth production (such as bribes to local police) then a reduction in the demand for moonshine will raise the costs of producing meth.

Understanding which of these theories holds would make an important contribution to the industrial organization of illegal goods and would also have implications for how best to combat illegal good production.

Open Borders and the Welfare State

by on September 22, 2015 at 7:22 am in Economics, Law | Permalink

Milton Friedman famously said that you can’t have a welfare state and open borders. I disagree. In many respects (not all), you can have open borders and a welfare state.

What we think of as the welfare state encompasses many different programs, many of which are not handouts. Social Security for example is mostly a forced savings program. For these types of insurance programs there is no problem at all as, for the most part, a person has to work and pay into the program to get money out of the program. For programs like schooling there is also no problem–even if the schooling is provided free to immigrant children–because the schooling leads to higher wages later in life which are taxed. In these cases, the immigrant children are really just receiving a loan which they will have to pay back from their own earnings later in life. The story for basic health is similar. Thus, the only cases where there is a worry about excessive transfers from citizens to immigrants is in pure handouts or health benefits to say the elderly. In these cases, I would simply say that such benefits are not available to immigrants or only available after five years or some such time period.

Addendum: I gave this answer in an interview for a Brazilian newspaper. You can read the full interview here although it is in Portuguese.

A Phool and His Money

by on September 21, 2015 at 7:21 am in Books, Economics | Permalink

I was disappointed with Akelof and Shiller’s Phishing for Phools.

Cinnabon pastries are hard to resist. Advertising can be deceptive. Humans sometimes act in foolish ways. If these statements strike you as anodyne then there is no need to read George Akerlof and Robert Shiller’s new book Phishing for Phools, a disappointing foray into behavioral economics from two recent Nobel Prize winners. If these statements strike you as novel then I recommend instead Ariely’s Predictably Irrational, Sunstein and Thaler’s Nudge, Kahneman’s Thinking, Fast and Slow or Daniel Gilbert’s Stumbling on Happiness, to name just a few classics in the field

You can read my full review at The New Rambler.

The econometrician Henri Theil once said “models are to be used but not to be believed.” I use the rational actor model for thinking about marginal changes but Gary Becker really believed the model. Once, at a dinner with Becker, I remarked that extreme punishment could lead to so much poverty and hatred that it could create blowback. Becker was having none of it. For every example that I raised of blowback, he responded with a demand for yet more punishment. We got into a heated argument. Jim Buchanan and Bryan Caplan approached from the other end of the table and joined in. BeckerIt was a memorable evening.

Becker isn’t here to defend himself on the particulars of that evening but you can see the idea in his great paper, Crime and Punishment: An Economic Approach. In a famous section he argues that an optimal punishment system would combine a low probability of being punished with a high level of punishment if caught:

If the supply of offenses depended only on pf—offenders were risk neutral — a reduction in p “compensated” by an equal percentage increase in f would leave unchanged pf… increased probability of conviction obviously absorbs public and private resources in the form of more policemen, judges, juries, and so forth. Consequently, a “compensated” reduction in this probability obviously reduces expenditures on combating crime, and, since the expected punishment is unchanged, there is no “obvious” offsetting increase in either the amount of damages or the cost of punishments. The result can easily be continuous political pressure to keep police and other expenditures relatively low and to compensate by meting out strong punishments to those convicted.

We have now tried that experiment and it didn’t work. Beginning in the 1980s we dramatically increased the punishment for crime in the United States but we did so more by increasing sentence length than by increasing the probability of being punished. In theory, this should have reduced crime, reduced the costs of crime control and led to fewer people in prison. In practice, crime rose and then fell mostly for reasons other than imprisonment. Most spectacularly, the experiment with greater punishment led to more spending on crime control and many more people in prison.

Why did the experiment fail? Longer sentences didn’t reduce crime as much as expected because criminals aren’t good at thinking about the future; criminal types have problems forecasting and they have difficulty regulating their emotions and controlling their impulses. In the heat of the moment, the threat of future punishment vanishes from the calculus of decision. Thus, rather than deterring (much) crime, longer sentences simply filled the prisons. As if that weren’t bad enough, by exposing more people to criminal peers and by making it increasingly difficult for felons to reintegrate into civil society, longer sentences increased recidivism.

Instead of thinking about criminals as rational actors, we should think about criminals as children. In this light, consider the “Becker approach” to parenting. Punishing children is costly so to reduce that cost, ignore a child’s bad behavior most of the time but when it’s most convenient give the kid a really good spanking or put them in time out for a very long time. Of course, this approach leads to disaster–indeed, it’s precisely this approach that leads to criminality in later life.

So what is the recommended parenting approach? I don’t want to get into a debate over spanking, timeouts, and reasoning but one thing all recommendations have in common is that the consequences for inappropriate behavior should be be quick, clear, and consistent. Quick responses help not just because children have “high discount rates” (better thought of as difficulty integrating their future selves into a consistent whole but “high discount rates” will do as short hand) but even more importantly because a quick response helps children to understand the relationship between behavior and consequence. Prior to Becker there was Becaaria and in Beccarian theory, people must learn to associate crime with punishment. When responses aren’t quick, children, just like scientists, have difficulty learning cause and effect. Quick is thus one way of lowering cognitive demands and making consequences clear.

Animals can learn via conditioning but people can do much better. If you punish the child who steals cookies you get less cookie stealing but what about donuts or cake? The child who understands the why of punishment can forecast consequences in novel circumstances. Thus, consequences can also be made clear with explanation and reasoning. Finally, consistent punishment, like quick punishment, improves learning and understanding by reducing cognitive load.

Quick, clear and consistent also works in controlling crime. It’s not a coincidence that the same approach works for parenting and crime control because the problems are largely the same. Moreover, in both domains quick, clear and consistent punishment need not be severe.

In the economic theory, crime is in a criminal’s interest. Both conservatives and liberals accepted this premise. Conservatives argued that we needed more punishment to raise the cost so high that crime was no longer in a criminal’s interest. Liberals argued that we needed more jobs to raise the opportunity cost so high that crime was no longer in a criminal’s interest. But is crime always done out of interest? The rational actor model fits burglary, pick-pocketing and insider trading but lots of crime–including vandalism, arson, bar fights and many assaults–aren’t motivated by economic gain and perhaps not by any rational interest.

Here’s a simple test for whether crime is in a person’s rational interest. In the economic theory if you give people more time to think carefully about their actions you will on average get no change in crime (sometimes careful thinking will cause people to do less crime but sometimes it will cause them to do more). In the criminal as poorly-socialized-child theory, in contrast, crime is often not in a person’s interest but instead is a spur of the moment mistake. Thus, even a small opportunity to reflect and consider will result in less crime. As one counselor at a juvenile detention center put it:

20 percent of our residents are criminals, they just need to be locked up. But the other 80 percent, I always tell them – if I could give them back just ten minutes of their lives, most of them wouldn’t be here.

ThinkingProblemsCognitive behavioral therapy teaches people how to act in those 10 minutes–CBT is not quite as simple as teaching people to count to ten before lashing out but it’s similar in spirit, basically teaching people to think before acting and to revise some of their assumptions to be more appropriate to the situation. Randomized controlled trials and meta-studies demonstrate that CBT can dramatically reduce crime.

Cognitive behavioral therapy runs the risk of being labeled a soft, liberal approach but it can also be thought of as remedial parenting which should improve understanding and appreciation among conservatives. More generally, it’s important that crime policy not be forced into a single dimension running from liberal to conservative, soft to tough. Policing and prisons, for example, are often lumped together and placed on this single, soft to tough dimension when in fact the two policies are different. I favor more police on the street to make punishment more quick, clear, and consistent. I would be much happier with more police on the street, however, if that policy was combined with an end to the “war on drugs”, shorter sentences, and an end to brutal post-prison policies that exclude millions of citizens from voting, housing, and jobs.

Let’s give Becker and the rational choice theory its due. When Becker first wrote many criminologists were flat out denying that punishment deterred. As late as 1994, for example, the noted criminologist David Bayley could write:

The police do not prevent crime. This is one of the best kept secrets of modern life. Experts know it, the police know it, but the public does not know it. Yet the police pretend that they are society’s best defense against crime. This is a myth

Inspired by Becker, a large, credible, empirical literature–including my own work on police (and prisons)–has demonstrated that this is no myth, the police deter. Score one for rational choice theory. It’s a far cry, however, from police deter to twenty years in prison deters twice as much as ten years in prison. The rational choice theory was pushed beyond its limits and in so doing not only was punishment pushed too far we also lost sight of alternative policies that could reduce crime without the social disruption and injustice caused by mass incarceration.

Japan is liberalizing its approval process for regenerative medicine:

…Regenerative medicines in Japan can now get conditional marketing approval based on results from mid-stage, or Phase II, human trials that demonstrate safety and probable efficacy. Once lagging behind the United States and the European Union on approval times, there is now an approximately three-year trajectory for approvals, according to Frost’s Kumar. That compares with seven to 10 years before.

…Around the world, companies have also faced setbacks while pushing such treatments. In the U.S., Geron Corp., which started the first nation-approved trial of human embryonic stem cells, ended the program in 2011, citing research costs and regulatory complexities.

…While scientists globally have worked for years in this field, treatments have been slow to come to market. But there is hope in Japan that without the political red tape, promising therapies will emerge faster and there will be speedier rewards.

Japan is liberalizing because with their aging population treatments for diseases like Alzheimer’s and Parkinson’s disease are in high demand. Under the new system, a firm with a gene or regenerative therapy (e.g. stem cells) can get conditional approval with a small trial. Conditional approval means that the firm will be able to sell its procedure while continuing to gather data on efficacy for a period of up to seven years. At the end of the seven year period, the firm must either apply for final marketing approval or withdraw the product. The system is thus similar to what Bart Madden proposed for pharmaceuticals in Free to Choose Medicine.

Due to its size and lack of price controls, the US pharmaceutical market is the most lucrative pharmaceutical market in the world. Unfortunately, this also means that the US FDA has an outsize influence on total world investment. The Japanese market is large enough, however, that a liberalized approval process if combined with a liberalized payment model could increase total world R&D.

Breakthroughs made in Japan will be available for the entire world so we should all applaud this important liberalization.

Hat tip: Michael Mandel.

Drum Solo

by on September 13, 2015 at 7:25 am in Music, The Arts | Permalink

Rush’s Neil Peart was recently voted the greatest drummer of all time. Here’s Neil demonstrating why:

To appreciate the artistry, I like this best on headphones without visuals but the video captures the amazing physicality of the performance. Some backstory here.

Gun Drop Boxes???

by on September 10, 2015 at 7:54 am in Economics | Permalink

Reporters often call me for insight into various issues or just a quotable comment. Yesterday, however, I had a hard time believing the policy the reporter wanted to talk about was real.

Tacoma, Washington wants to install boxes around the city so people can drop off their guns, no questions asked. But, but, but…seriously?

After the reporter assured me this was real I offered the following:

George Mason University economics professor Alex Tabarrok said [gun buyback] programs have been proven ineffective, and he predicted the same result for a drop box scheme. Anyone in the port city who wants to get rid of a gun discreetly, Tabarrok said, has an easy option: toss it in Puget Sound.

Not really something you need a PhD for but when you are tossed a softball you may as well hit it.

There is something to be said for simply reducing the hassle of disposing a gun for those who don’t want to sell but as a crime control measure this is a bust. Regardless of your views on guns not much will change if you don’t do something nationally about the big boxes where people can buy guns (otherwise known as stores).

On Labor Day let’s all read the excellent White House report on occupational licensing:

…the current licensing regime in the United States also creates substantial costs, and often the requirements for obtaining a license are not in sync with the skills needed for the job. There is evidence that licensing requirements raise the price of goods and services, restrict employment opportunities, and make it more difficult for workers to take their skills across State lines. Too often, policymakers do not carefully weigh these costs and benefits when making decisions about whether or how to regulate a profession through licensing. In some cases, alternative forms of occupational regulation, such as State certification, may offer a better balance between consumer protections and flexibility for workers.

I recently ran across a problem using the Frechet probability bounds. The bounds weren’t immediately obvious to me and Google didn’t enlighten so I wrote an intuitive explanation. Super wonkiness to follow. You have been warned.

Suppose that we have two events A and B which occur with probability P(A) and P(B). We are interested in saying something about the joint event P(A∩B), the probability that both events occur, but we don’t know anything about whether the events are correlated or independent. Can we nevertheless say something about the joint event P(A∩B)? We can. The Frechet bounds or inequalities tell us:

max[0,P(A)+P(B)-1] ≤ P(A∩B) ≤ min[P(A),P(B)]

In words, the probability of the joint event can’t be smaller than max[0,P(A)+P(B)-1] or bigger than min[P(A),P(B)]. Let’s give an intuitive explanation.

The events themselves are not important only the probabilities so let’s use an intuitive model for the events. Let x be a random number distributed between 0 and 1 with each number equally likely (i.e. a uniform distribution). Suppose that the event A occurs whenever x is in some region between a and b. For example, we might say that event A occurs if x is between .4 and .6 and event B occurs if x is between .1 and .7. Notice also that since any number between 0 and 1 is equally likely the probability that event A occurs is just the width of the A region, b-a. In this case, for example, P(A)=.6-.4=.2 and similarly P(B)=.7-.1=.6.

Now suppose that a demon arranges the events A and B to minimize the probability that both events occur. What is the smallest the demon could make P(A∩B)? To minimize the probability that both A and B occur we can think of the demon as placing the two events “as far apart” as possible. For example, the demon will begin one event at 0 and define it as the region moving right until the width is equal to P(A) and the other event the demon will begin at 1 and move left until the width is equal to P(B). If P(A)=.2 and P(B)=.4, for example, then the demon will define the events so that A∈[0,.2] and B∈[.6,1]. Here’s a picture. Notice that by beginning one event at 0 and the other at 1 the demon minimizes the overlap which is the probabiity that both events occur.

Frechet1In this case, the events don’t overlap at all and so the demon has ensured that the probability of P(A∩B)=0.

But what if the two events must overlap? Suppose for example that P(A)=.5 and P(B)=.7 then the demon’s logic of minimization leads to the following picture:

Frechet2We can see from the picture that the overlap is the region between .3 and .5 which we know has probability .5-.3=0.2. Thus we have discovered that if P(A)=.5 and P(B)=.7 then the joint probability has to be at least .2, i.e. .2≤P(A∩B).

Now let’s generalize. The probability of any event occuring is the length of its region b-a but since the demon always begins the event A at 0 then the probability of P(A) is simply b the rightmost point of the A region. Thus, on the diagram the rightmost point of the A region is labelled P(A) (.5 in this case). Also since the demon always begins the B region at 1, P(B)=1-a′ where a′ is the leftmost point of the B region so rearranging we have that the leftmost point a′=1-P(B) as shown in the picture at .3. Thus, we can read immediately from the figure that the overlap region has length P(A)-(1-P(B)). Notice that if P(A)<1-P(B) then there is no overlap and the difference is a negative number. Rearranging slightly, the length of the overlap region is P(A)+P(B)-1. So now we have two cases, either there is no overlap at all in which case P(A∩B)=0 or there is overlap and P(A∩B)=P(A)+P(B)-1. So putting it all together we have proven that:

max[0,P(A)+P(B)-1] ≤ P(A∩B)

The Frechet bounds also say:

P(A∩B) ≤ min[P(A),P(B)]

This is much easier to show. If minimization is accomplished by minimizing the overlap then maximization is accomplished by maximizing the overlap. To maximize the joint probability the demon starts both regions at the same point, say at the 0 point, so the picture looks like this:
Frechet3The joint probability is then the region of overlap. But the regions can’t overlap by more than the smallest of the two regions! In this case the A region is smaller than the B region and since the regions start at 0 we have P(A∩B)≤P(A). Generalizing we have that:

P(A∩B) ≤ min[P(A),P(B)]

Thus we have proven the Frechet bounds:

max[0,P(A)+P(B)-1] ≤ P(A∩B) ≤ min[P(A),P(B)]

Addendum: Here is Heckman on the theory of Frechet Bounds and Heckman and Smith and Heckman, Smith and Clements with applications.