The Faller Curve : On a Permutation of Consonants and Objectives

English people like to live in individual one-floor-high English houses with English gardens in 50-English-unit-of-measurement-wide streets. It is hard to see what they get out of that, but never ending cities, expensive real estate, and 2 hours trips in the suburbs before being able to reach a highway. It is easier to undersand what this blog can gain from it.

For the occurence of trips in buses creates a craving for anything in a position to occupy the mind which is more easily satisfied in urban landscape. For the sake of clearness, we will restrict this analysis to blue walls with peaks. Indeed, somewhere in the londonian suburbs, there is a blue blank wall in which a bunch of horizontal landings are embedded. This wall could therefore easily be climbed, hadn't the city council (or anybody else) display peaks on each horizontal surface. We can assume that the peaks are there in order to deter people from climbing to the top (although we cannot exclude other explanations) and avoid the accidents of people falling. The peak strategy is of course understandable, but, on the other hand, such peaks make the climbing more difficult, hence increase the probability of falling (if not the damage of one fall, but in order to have a clear measurement of the danger, we will restrict ourselves to the probability of falling). So should the city council put peaks on its wall? And if so, how many?

A easy way to adress the problem is to recognize its price-quantity dilemma nature.

Indeed, the probability of falling can be thought of as a price in reaction to which people on the demand-for-wall-climbing side make their decision of climbing the wall. We will assume that climbing walls is fun, but that people do not value the climbing equally, and do not arbitrate between the risk of falling and the pleasure of climbing in the same way, depending on their risk aversion and their wall tastes. We will also assume that the decision to climb the wall is individually decreasing with its price/probability of falling (no challenge effect or, equivalently, climbing wall is not a luxury good). We then have a strictly decreasing demand function.

The number of falls, provided a sufficient number of people to apply the law of large numbers, is then the probability of falling times the number of persons climbing the wall, which can be represented graphically by the area of a rectangle.

Let us assume now that the information is perfect, so that the city council can perfectly set the probability of a fall, for instance by displaying peaks on the wall, or by any other mean available (including means which would decrease the probability by increasing safety), and that the climbers can accurately and with no mistake deduce the real danger (assimilated to the probability of falling) from the view of the wall.

The problem becomes easily assimilated to a choice of price by a monopoly, or the choice of a tax rate by a government. Except that if the city council now assigns itself an objective of minimization of the falls, the objective is in this analogy to minimize the profit, or the revenue of taxation. The graphic of the problem will therefore lead to a Laffer Curve of a particular type, namely a Faller Curve, displayed below.

So how dangerous should the city council make the climbing? The answer is quite intuitive: the most dangerous they can, in order to deter everyone from climbing (or more rigorously dangerous enough so that the marginal peak will have no impact on the will to climb) or the least they can. Exactly like a monopoly would be sure to make no profit by choosing an infinite price or by giving its production away for free.

Contrary to the maximization problem, and assuming that neither a riskless wall nor a wall no one would ever want to climb is possible, the city council faces a binary choice, between the riskest and the safest walls available. In order to choose between those two solutions, it needs to have some information about the risk elasticity of the citizens, as in the usual Laffer Curve. If citizens are bold enough so whatever the danger, they will climb, the city hall may want to choose to improve safety up to what is possible. On the other hand, if people are very uncomfortable with the risk of falling, the minimizing solution will be to display as many peaks as possible.

Besides, we assumed incresing or decreasing safety was costless, but safety improvements at least, if not peak display, are likely to be costly. Which side of the curve the wall is initially on before any council intervention would then matter for the choice of the solution, as in the usual Laffer Curve.

Let us conclude by noticing that the maximisation of the surplus of the climbers would still lead to the minimization of the probability up to what is possible, even if this means more chance of falling and more aggregate falls, just as consumers are happier when goods are the least expensive : since we assumed perfect information, people would climb knowing the risk they take in doing so. The objective of the city council to minimize the number of falls is therefore questionable, as long as we assume perfect information and perfect knowledge of the risk by the climbers. So the problem revolves without surprise around the legitimacy of city council paternalism.

Let us not be too enthusiastic though. This analysis tells us nothing about how to spend the 6 hours left.

EDIT: I shall remind that the effect described in this post is traditionally referred to under a much duller name, thanks to Mr. Peltzman, who found his own appropriate enough.

A post about nothing

A Option Value Problem from Seinfled, dixit Dixit.

A little reminder if needed.

"Correlation is not causation, even in the blogosphere"

It is by Gregory Mankiw, and it was posted today.

Morality : sometimes, temporal succession not being causation sucks.

The Protestant Ethic and the Spirit of Statistics

The readers of the previous post may be of the kind who is very reluctant to accept an history of anything which does not make a large case for protestantism. Fortunately, this was in our case a mere omission. For we are strong defenders of the idea that, like most of the time, the answers to most of our questions are to be found in calvinism.

It is a common defect of historians to be unable to see the statistical nature of most of the historical controversies they study. The Reformation's case is a good example of such an inability. Indeed, religious aspects of the emergence of calvinism should not hide what must be considered as one of the first attempts to state the difference between correlation and causation. For here is the primary message of calvinism: the Roman Church, the Calvinists say, noting that virtuous lives are strongly correlated with an afterlife in paradise, jumped to the conclusion that the virtuous actions cause their author to go to Heaven. This is a poor understanding of the statistical nature of this correlation. To the contrary, we can bring strong evidences that a common-causal variable, namely "being chosen", causes both virtuous life and acceptance to Heaven.

And this is the kind of change that can account for the European economic take-off.

To conlcude this opening lecture on the history of statsistics, we shall notify that the two precedent posts are inspired by Quattrone & Tversky's 1984 article, Causal versus diagnostic contingencies: On self-deception and on the voter's illusion, a summary of which can be found here. (Tversky was a cognitive psychologist, co-author for 10 years of 2002 Economics Nobel Prize-recipient Daniel Kahneman, with who he conceived the Prospect Theory, the most famous non Expected-Utility theory of decision in uncertainty. It is commonly accepted that he would have jointly received the prize, hadn't he died in 1996.)

Tversky and Quattrone's analysis is opposite to this one : according to them, Calvinists would behave as if their actions were causes of where they will go in the afterlife, even though their doctrine asserts that the correlation between virtuous lives and Heaven is by no mean causal. More precisely, they take Calvinists as an example of deceptive diagnosis, which is a form of self-deception where people select an action to make a favorable diagnosis of a positive outcome as if the favorable diagnosis caused the positive outcome : people deceive themselves by mistaking the correlation between the favorable diagnosis and the positive outcome for a causal relation. A calvinist tempted soul could for instance resist the temptation, even though his temptation is already the proof that he is not chosen ("Calvinists ought to transgress upon experiencing the desire to do so, other things being equal, for heaven and hell are independant of actions conditional to the urge to sin. Contrary to this analysis, we believe that many Calvinists would nonetheless resist the temptation and choose instead the virtuous acts correlated with paradise."). Interestingly, for Weber's argument on the origins of accumulation in calvinism to be valid, we need to assume the existence of such kind of self-deceiving process, which is confirmed by Tversy and Quattrone's experiment.

Fisher's Chest: An Inquiry into the Nature of Causes, Health and Caution

Man has always been learning that correlation does not imply causation. When expulsed from the Garden of Eden, Adam had acquired from the Tree of Knowledge the understanding that he could by no mean assert that the temporal correlation between his apple biting and his expulsion from the Garden of Eden was a causal one. Since then, each time and civilisation has developped its own way to state and teach the principle, so much that the evolution of this instruction can be seen as a satisfactory proxy for the study of human history. It is therefore difficult to understand why so little attention has been paid so far for making an history of the teaching of the primary principle of statistics. It was in the attributes of this blog to fill the gap.

What causes people to discover that correlation does not imply causation? The generation to which I failed to belong learnt it the hard way. As for the second half of the XXth century, most people were then taught that being black was not necessarily the cause of poverty, since one could not reject the reverse causality, poverty turning people black. At the beginning of the XXth century, Professor Udny Yule told the story of a large significant correlation between the number of apples imported into Great Britain and the number of divorces.

This last exemple is cited by Ronald Fisher, probably the most important statistician of the XXth century, whose contributions to science seem however to have been highly selected in the freshman statistics handbooks. For behind the test is an incorruptible man of principles, whose fights for science have been kept in silence. He is for instance the author of one of the best attempts to teach the public opinion with the difference between correlation and causality. His major contributions to the subject are gathered in his book Smoking, The Cancer Controversy, Some Attempts to assess the Evidence, which gathers many of his articles, most of them written as a reaction to the conclusions of Richard Doll and A.B. Hill on lung cancer, from which the passage below is taken :

The subject is complicated, and I mentioned at an early stage that the logical distinction was between A causing B, B causing A, something else causing both. Is it possible, then, that lung cancer – that is to say, the pre-cancerous condition which must exist and is known to exist for years in those who are going to show overt lung cancer – is one of the causes of smoking cigarettes? I don’t think it can be excluded. I don’t know enough to say it is such a cause. But the pre-cancerous condition is one involving a certain amount of slight chronic inflammation. The causes of smoking cigarettes may be studied among your friends, to some extent, and I think you will agree that a slight cause of irritation – a slight disappointment, an unexpected delay, some sort of a mild rebuff, a frustration – are commonly accompanied by pulling out a cigarette and getting a little compensation for life’s minor ills in that way. And so, anyone suffering from a chronic inflammation in part of the body (something that does not give rise to conscious pain) is not unlikely to be associated with smoking more frequently, or smoking rather than not smoking. It is the kind of comfort that might be a real solace to anyone in the fifteen years of approaching lung cancer. And to take the poor chap’s cigarettes away from him would be rather like taking away his white stick from a blind man. It would make an already unhappy person a little more unhappy than he need be.

Fisher's book is available online, in particular this paper, whose title would be thought to be borrowed from a Jim Jarmusch's movie, if we did not know it was the reverse causation at play.

The outkirst of Bruges lost in the sea

Who leaves a country packed with ponies to come to a non-pony country?

Jerry Seinfeld

We might have been so obsessed with endings that we forgot to say how things always begin. Worse, we forgot all about chronology. But since it is so much easier to forget one's past than his future, this blog will not look back in anger, you heard me say.

Econometrics and the linen: from fitted values to the value of fitted sheets

"It is best to wash one's linen at home" : it follows that the best way to investigate the culture of a nation is to study how it deals with its supply of linen. The inextricable network of habits which defines the culture of a country and marks its contrasts with the others has been known since 1994 as "the little differences". It was shown at the time that, as far as anglo-saxon countries are concerned, most of those little differences exist "because of the metric system". But how does the story fit when jumping from Quarter Pound with Cheese to fitted sheet?

At first sight, beds, mattresses, covers and sheets in the English-speaking world seem to get in the metrics. Bedding and linen are there ranked between Single, the same 90 cm wide beds than on the continent and Double, the 140 cm wide beds most continental couples are used to. But as soon as one wants a wider bed, he has to get out of the metrics : beds are then " 5 feet wide" or "6 feet wide". The Empire has given in on the smallest beds, but not on the biggest ones. It tells us a lot about the diffusion of the metric system among fat people in the UK.

This analysis, combined to the study of the natality curves in the United Kingdom, can be used to predict the forthcoming collapse of the British Empire, and gives us the right to say whatever we want about economics for the years to come.

The linen case could actually be generalized to all the bedding field. It is not what this post is primarily dealing with, but we would then come to the conclusion that bedding is extremely exposed to lath dependency.