prior probability

Thus far this week (see here and here), we have pored over two intriguing selections from The Wealth of Nations — intriguing because Smith refers to the concepts of probability and risk in both of those passages in order to shed light on two different aspects of the economy. (One is a short paragraph in Book V, Chapter 1 of Smith’s magnum opus on the business of insurance. The other passage consists of a much longer paragraph in Book IV, Chapter 2, where the Scottish scholar is discussing the imposition of retaliatory tariffs.) But, for me, the most original and profound of these probability passages in The Wealth of Nations appears in paragraphs 22 and 23 of Book I, Chapter 10. Here, Smith explains how, on average, one’s wages are a function of the probability of success in one’s chosen profession:

“The probability that any particular person shall ever be qualified for the employment to which he is educated is very different in different occupations. In the greater part of mechanic trades, success is almost certain; but very uncertain in the liberal professions. Put your son apprentice to a shoemaker, there is little doubt of his learning to make a pair of shoes; but send him to study the law, it is at least twenty to one if ever he makes such proficiency as will enable him to live by the business. In a perfectly fair lottery, those who draw the prizes ought to gain all that is lost by those who draw the blanks. In a profession where twenty fail for one that succeeds, that one ought to gain all that should have been gained by the unsuccessful twenty. The counsellor-at-law who, perhaps, at near forty years of age, begins to make something by his profession, ought to receive the retribution, not only of his own so tedious and expensive education, but that of more than twenty others who are never likely to make anything by it. How extravagant soever the fees of counsellors-at-law may sometimes appear, their real retribution is never equal to this. Compute in any particular place what is likely to be annually gained, and what is likely to be annually spent, by all the different workmen in any common trade, such as that of shoemakers or weavers, and you will find that the former sum will generally exceed the latter. But make the same computation with regard to all the counsellors and students of law, in all the different inns of court, and you will find that their annual gains bear but a very small proportion to their annual expense, even though you rate the former as high, and the latter as low, as can well be done. The lottery of the law, therefore, is very far from being a perfectly fair lottery; and that, as well as many other liberal and honourable professions, are, in point of pecuniary gain, evidently under-recompensed.” (WN, I.x.b.22, pp. 122-123)

In other words, to quote Smith, “the wages of labour in different employments vary according to the probability or improbability of success in them.” (WN, I.x.b.21, p. 122) Shoemaking, to borrow Smith’s example, is a low-risk profession in which “success is almost certain”. Why? Because anyone, no matter how slow-witted, can learn to make a decent pair of shoes with enough practice and training. As a result, although most shoemakers will make enough money to scrape by, none will become super-rich because low-risk professions produce, at best, average-level rewards.

The legal profession, however, is a high-risk profession. Why? Because getting a law degree is expensive in terms of both time and money, since it takes many years of formal studies to become a licensed attorney, and because most new attorneys are doomed to fail. According to Smith, out of a random sample of 21 lawyers, 95% (20 divided by 21) of them will end up scraping by, unable to make a good living in their chosen profession. Given these probabilities, doesn’t it make more sense to make shoes? So, why do so many young people still want to become lawyers? Smith explains why in the next paragraph (para. 23) of Book I, Chapter 10:

“Those professions keep their level, however, with other occupations, and, notwithstanding these discouragements, all the most generous and liberal spirits are eager to crowd into them. Two different causes contribute to recommend them. First, the desire of the reputation which attends upon superior excellence in any of them; and, secondly, the natural confidence which every man has more or less, not only in his own abilities, but in his own good fortune.” (WN, I.x.b.23, p. 123)

Smith thus identifies two variables to explain why lawyers outnumber shoemakers, even though the probability of success is much lower for lawyers than it is for shoemakers. One variable is the level of “public admiration” that a given profession has (I.x.b.24, p. 123), or as Smith more plainly puts it: “the desire of … reputation”. (I.x.b.23, p. 123) A lot of people want to join the legal profession because people generally admire lawyers more than they do shoemakers. (As an aside, Smith’s analysis of wage differentials in many ways sets the stage for Sherwin Rosen’s much-cited paper “The Economics of Superstars” (Rosen 1981), available here. See also paragraph 25 of Book I, Chapter 10 of The Wealth of Nations, where Smith explores the other side of the coin: those professions, like exotic dancers, that are considered a form of public prostitution.)

The other crucial variable is self-deception, or in the immortal words of Adam Smith, “the natural confidence which every man has more or less, not only in his own abilities, but in his own good fortune.” (I.x.b.23, p. 123) I will say more about self-deception in my next post. For now, I just want to say that Smith’s point about the psychology of self-deception is especially provocative and ahead of its time. It not only anticipates Robert Triver’s cutting-edge work on self-deception (see, for example, his 2011 book below); it also rejects and refutes the standard “rational actor model” that has been taught in introductory economics courses for generations! (To be continued …)

My previous post explored the first of three passages in The Wealth of Nations in which Adam Smith explicitly refers to the concept of probability. Today, I want to discuss another such passage: the retaliatory tariff passage in Book IV, Chapter 2 of Smith’s magnum opus:

“There may be good policy in retaliations of this kind, when there is a probability that they will procure the repeal of the high duties or prohibitions complained of. The recovery of a great foreign market will generally more than compensate the transitory inconveniency of paying dearer during a short time for some sorts of goods. To judge whether such retaliations are likely to produce such an effect, does not, perhaps, belong so much to the science of a legislator, whose deliberations ought to be governed by general principles, which are always the same, as to the skill of that insidious and crafty animal vulgarly called a statesman or politician, whose councils are directed by the momentary fluctuations of affairs. When there is no probability that any such repeal can be procured, it seems a bad method of compensating the injury done to certain classes of our people, to do another injury ourselves, not only to those classes, but to almost all the other classes of them. for that alone would seldom affect them considerably, but some other manufacture of theirs.” (WN, IV.ii.39, p. 468, my emphasis)

In other words, whether country A should impose a retaliatory tariff on country B as a strategic device to induce the repeal of country B’s tariffs depends on the probability that such a retaliatory tariff will, in fact, succeed in persuading country B to repeal its tariffs. Common sense, right? No, it’s more than that, for Adam Smith is making two deeper points here. One is that law and politics is fundamentally a probabilistic game. Indeed, one of the most important lessons of Smith’s works is that new laws, no matter how well-intentioned, will always produce unforeseen consequences.

More importantly, this observation about the probabilistic nature of law and politics leads us directly to Adam Smith’s second major insight in the retailiatory tariff passage quoted above: Who is in the best position to calculate what the relevant probabilities are? Who decides? Also worth noting here is that — regardless of what we make of Smith’s distinction between skill and science, i.e. between “the skill of that insisdious and crafty animal vulgarly called a statesman or politician” and “the science of the legislator”, to be more precise — Smith’s “who question” is, at bottom, a normative question: Who should decide?

This insight, in turn, brings me back to my days as a young assistant professor of constitutional law at the Pontifical Catholic University of Puerto Rico. When I began teaching “conlaw” in 1998, one of the law review articles that most influenced my thinking about my subject was “Who Decides?” by Paul Brest (pictured below), who I consider one of my intellectual mentors. Alas, I am unable to find an ungated or free version of this paper, so I will summarize it here. In brief, one of the big questions in constitutional law is, How should we interpret our nation’s founding charter? Does its meaning change over time (e.g. the “Living Constitution”), or is the meaning fixed and stable (e.g. Originalism)? Professor Brest, by contrast, flipped the script. He asked a totally different question: Who gets to decide what the meaning of the Constitution is? His main point was that judges are not the only interpreters. Members of the legislative and executive branch also have a non-delegable duty to interpret the Constitution when they are enacting and enforcing laws.

Now, isn’t Adam Smith saying the same thing about law and politics more generally? To sum Smith up: (1) the desired effects of new laws are probabilistic because of their unintended consequences, and more importantly, (2) it matters who gets to decide or calculate what these probabilites are in the first place?

Work cited: Paul Brest, Who decides?, Southern California Law Review, Vol. 58 (1985), pp. 661-672.

Happy Paris Liberation Day! (And Happy Birthday to yours truly!) In my previous post, I introduced and surveyed three separate passages from The Wealth of Nations in which Adam Smith refers in one way or another to the concept of probability. (Hat tip to my colleague and new friend Michael Emmett Brady for re-bringing these probability passages to my attention.)

Today, I will return to the first of these three Smithian selections: the insurance business paragraph. To begin, this particular passage, which appears for the first time [1] in the last part of a 79-page pamphlet Smith published in 1784 (see Smith 1784a, p. 77) and which was then inserted into Book V, Chapter 1 of the third edition of Smith’s magnum opus (Smith 1784b, Vol. 3, p. 147, available here), reads as follows:

“The value of the risk, either from fire, or from loss by sea, or by capture, though it cannot, perhaps, be calculated very exactly, admits, however, of such a gross estimation, as renders it, in some degree, reducible to strict rule and method. The trade of insurance, therefore, may be carried on successfully by a joint-stock company, without any exclusive privilege. Neither the London Assurance, nor the Royal Exchange Assurance companies have any such privilege.” (WN, V.ii.b.34, p. 756, my emphasis)

[1] The possible significance of the actual publication timeline of this short passage will become clear shortly.

One can certainly try to argue, as Professor Brady does, that Smith’s off-hand remark that “the risk … cannot … be calculated very exactly” anticipates the idea of “imprecise probabilities” (see here and here, for example), but I want to focus instead on the second part of the passage above, where Smith refers to two actual insurance companies: London Assurance and Royal Exchange Assurance. Historian Maurice Edward Ogborn reports that both of these companies were granted their corporate charters during the South Sea Bubble of 1720:

“In the troubled times of the South Sea Bubble, charters were granted to two corporations, the London Assurance and the Royal Exchange Assurance—in 1720 for the purpose of transacting marine insurance business and in 1721 for fire and life insurance business. The corporations granted life assurance contracts for a year at a time, similar to the contracts that were being granted for fire insurance business.” (Ogborn 2006, p. 21)

Although both the London Assurance and Royal Exchange Assurance companies “were permanent and substantial organizations” (ibid.), the fields of actuarial science and risk management — or what I like to call “applied probability” — were still in their infancy. Case in point: according to Ogborn, the schedule of premiums of the first life insurance policies ever issued by the London Assurance and the Royal Exchange Assurance companies “was at a flat rate for a wide range of ages”!!! (Ibid.)

Instead, it was another insurance company — alas, one that is not mentioned by Adam Smith, either in the passage above or anywhere else in The Wealth of Nations for that matter — that pioneered age-based premiums based on mortality rates and that laid “the framework for scientific insurance practice and development” and “the basis of modern life assurance upon which all life assurance schemes were subsequently based.” (See here.) That company was the Equitable Society:

“The original establishment of life assurance upon a sound basis was largely the achievement of one office and of the men who served her. This was the Society for Equitable Assurances on Lives and Survivorships, now known as The Equitable Life Assurance Society, and still affectionately called the ‘Old Equitable’. The Equitable was the first life assurance society to grant long-term contracts of life assurance for either a stated period or the whole of life, with premiums calculated according to age and type of assurance.” (Ogborn 2006, p. 19)

Moreover, in a plot twist that proves the old adage “truth is stranger than fiction”, one of the leading intellectual lights who served this company was none other than Adam Smith’s nemesis, Richard Price (whose portrait is pictured below). [2] In short, if the insurance business is now, to quote Adam Smith, “reducible to strict rule and method“, this happy state of affairs was in large part due to the ground-breaking work of Richard Price!!! (See generally Chapter 4 of Frame 2015 as well as Chapter 7 of Ogborn 2006.)

[2] On Adam Smith’s low opinion of Richard Price, see Smith’s letter of 22 December 1785 in The Correspondence of Adam Smith. I quote from that letter here.

Furthermore, it was most likely Richard Price’s early work on Bayes’s theory of inverse probability in 1763 that eventually led to his interest in life insurance. (See, e.g., Frame 2015, pp. 44-45; see also Ogborn 2006, p. 88) According to Ogborn (p. 90), “The earliest contacts between Price and the Equitable Society are recorded in copies of his correspondence between August 1768 and March 1771, and it was Price, applying Bayesian methods to life insurance, who “counselled that the Society, ought to have tables of values for all ages and every different term of years in order to guard against ‘careless or unskilful calculations’.” (Ibid.)

The Equitable Society was founded in 1762 — 14 years before the publication of the first edition of The Wealth of Nations and 22 years before the publication of Smith’s 1784 pamphlet — so her omission from Smith’s magnum opus is curious. Indeed, I would go as far as to say that this omission requires an explanation. After all, if this 1785 letter is any indication (see Corr. 251), Adam Smith knew who Richard Price was and both men ran in same social circles during their visits to London.

WORKS CITED:

Frame, Paul. Liberty’s Apostle: Richard Price, His Life and Times. University of Wales Press (2015).

Ogborn, Maurice Edward. Equitable Assurances: The Story of Life Assurance in the Experience of the Equitable Life Assurance Society, 1762-1962. Routledge (2006).

Smith, Adam. An Inquiry into the Nature and Causes of the Wealth of Nations (R. H. Campbell and A. S. Skinner, editors), 2 vols. Liberty Fund (1981).

The Correspondence of Adam Smith. (Mossner, E. C., and Ross, I. S., editors). Liberty Fund (1987).

In a previous post, I asked whether Adam Smith was a closet Bayesian, and I concluded that post with a reference to an obscure but intriguing paper titled “A study of Adam Smith’s original contributions to economic theory and decision making under uncertainty” by Michael Emmett Brady, a lecturer at the Dominguez Hills campus California State University. (See also his 2015 book of essays on Adam Smith and John Maynard Keynes below.)

Among other things, Brady’s 2016 paper identifies three passages in The Wealth of Nations as proof of “Adam Smith’s original contributions to economic theory and decision making under uncertainty.” (Brady 2016, p. 39) For reference, I reproduce the three relevant passages below:

PASSAGE #1:

The first passage Professor Brady cites is from Book V, Chapter 2 of The Wealth of Nations, where Smith is explaining why the business of insurance can be successfully carried out by a joint stock company in a free market, i.e. without monopoly rights:

“The value of the risk, either from fire, or from loss by sea, or by capture, though it cannot, perhaps, be calculated very exactly, admits, however, of such a gross estimation, as renders it, in some degree, reducible to strict rule and method. The trade of insurance, therefore, may be carried on successfully by a joint-stock company, without any exclusive privilege. Neither the London Assurance, nor the Royal Exchange Assurance companies have any such privilege.” (WN, V.ii.b.34, p. 756; Brady’s emphasis)

For Prof. Brady, this passage is evidence in support of the proposition that “probabilities are not precise.” This may not sound like a big deal, but it really is. See here and here, for example.

PASSAGE #2:

The next probability-themed passage Prof. Brady mentions is an extended one that is spread across three paragraphs in Book I, Chapter 10 of The Wealth of Nations, where Smith is explaining “inequalities” in the wages of labour:

“… the wages of labour in different employments vary according to the probability or improbability of success in them.

“The probability that any particular person shall ever be qualified for the employment to which he is educated is very different in different occupations. In the greater part of mechanic trades, success is almost certain; but very uncertain in the liberal professions. Put your son apprentice to a shoemaker, there is little doubt of his learning to make a pair of shoes; but send him to study the law, it is at least twenty to one if ever he makes such proficiency as will enable him to live by the business. In a perfectly fair lottery, those who draw the prizes ought to gain all that is lost by those who draw the blanks. In a profession where twenty fail for one that succeeds, that one ought to gain all that should have been gained by the unsuccessful twenty. The counsellor-at-law who, perhaps, at near forty years of age, begins to make something by his profession, ought to receive the retribution, not only of his own so tedious and expensive education, but that of more than twenty others who are never likely to make anything by it. How extravagant soever the fees of counsellors-at-law may sometimes appear, their real retribution is never equal to this. Compute in any particular place what is likely to be annually gained, and what is likely to be annually spent, by all the different workmen in any common trade, such as that of shoemakers or weavers, and you will find that the former sum will generally exceed the latter. But make the same computation with regard to all the counsellors and students of law, in all the different inns of court, and you will find that their annual gains bear but a very small proportion to their annual expense, even though you rate the former as high, and the latter as low, as can well be done. The lottery of the law, therefore, is very far from being a perfectly fair lottery; and that, as well as many other liberal and honourable professions, are, in point of pecuniary gain, evidently under-recompensed.

“Those professions keep their level, however, with other occupations, and, notwithstanding these discouragements, all the most generous and liberal spirits are eager to crowd into them. Two different causes contribute to recommend them. First, the desire of the reputation which attends upon superior excellence in any of them; and, secondly, the natural confidence which every man has more or less, not only in his own abilities, but in his own good fortune.” (WN, I.x.b.21-23, pp. 122-123; Brady’s emphases)

For Brady, the above passage illustrates the problem of “unreliable probabilities with different weights of evidential support.”

PASSAGE #3:

Lastly, Brady refers to Adam Smith’s discussion of retaliatory tariffs in Book IV, Chapter 2 of The Wealth of Nations. According to Smith, whether country A should impose a retaliatory tariff on country B as a strategic device to induce the repeal of country B’s tariffs depends on the probability that such a retaliatory tariff will, in fact, succeed in persuading country B to repeal its tariffs. But who is in the best position to make such a probability calculus? Smith tells us in Book IV, Chapter 2:

“There may be good policy in retaliations of this kind, when there is a probability that they will procure the repeal of the high duties or prohibitions complained of. The recovery of a great foreign market will generally more than compensate the transitory inconveniency of paying dearer during a short time for some sorts of goods. To judge whether such retaliations are likely to produce such an effect, does not, perhaps, belong so much to the science of a legislator, whose deliberations ought to be governed by general principles, which are always the same, as to the skill of that insidious and crafty animal vulgarly called a statesman or politician, whose councils are directed by the momentary fluctuations of affairs. When there is no probability that any such repeal can be procured, it seems a bad method of compensating the injury done to certain classes of our people, to do another injury ourselves, not only to those classes, but to almost all the other classes of them. for that alone would seldom affect them considerably, but some other manufacture of theirs.” (WN, IV.ii.39, p. 468; no emphases by Brady)

QUESTIONS:

Do these three passages really amount to “original contributions” to decision making under uncertainty? Can a case be made that Adam Smith was a closet Bayesian? And, lastly (for now), why doesn’t Professor Brady cite Smith’s more extensive and prescient discussion of the psychology of lotteries and “the certainty or uncertainty of the returns” in Book I, Chapter 10 of The Wealth of Nations, especially paragraphs 27 to 33? I will return to and further discuss all of these questions in my next few posts later this week.

WORKS CITED:

Brady, Michael Emmett. 2015. Adam Smith: Essays on Adam Smith, John Maynard Keynes, and Their Interval Valued Approaches to Probability, Decision Making, and Uncertainty. Xlibris.

Brady, Michael Emmett. 2016. A study of Adam Smith’s original contributions to economic theory and decision making under uncertainty. International Journal of Business Policy & Governance, 3(3): 39-50, https://thescholedge.org/index.php/sijbpg/article/view/290

Smith, Adam. 1981. An Inquiry into the Nature and Causes of the Wealth of Nations (R. H. Campbell and A. S. Skinner, editors), 2 vols. Liberty Fund.

I was listening to a rebroadcast of an old recording of American Top Forty with the legendary Casey Kasem on my second-favorite SiriusXM channel — 70s on 7 — the other day when this oldie by the musical duo Pratt & McClain popped up. Their hit song, which peaked at No. 5 on the AT40 in 1976, is still one of my favorites from my childhood days!

By all accounts (see, e.g., Frame 2015, p. 44; see also here and here), the Rev. Thomas Bayes (1701-1761) developed his now-famous theorem in the 1740s but never published his remarkable work during his lifetime; instead, it was his close friend and fellow dissenting minister, the moral philosopher and mathematician Richard Price (1723-1791), who edited and communicated Bayes’s unpublished work via a letter to physicist John Canton (1718-1772), who then read Bayes’s paper, along with an appendix prepared by Price, aloud to the famed Royal Society of London on 23 December 1763. (The mathematician-astronomer Pierre-Simon Laplace (1749-1827) made additional contributions to Bayesian probability in the late 1700s; see here.)

For his part, Adam Smith’s familiarity with inverse probability is unclear at best. Although Smith was elected a Fellow of the Royal Society on 21 May 1767, he could not have been present at that 1763 meeting of the Royal Society in which Bayes and Price’s work on probability was discussed (Smith was still in Glasgow), and he was either en route to France or in France when Bayes’s original paper and Price’s appendix were finally published and circulated (Bayes 1763; Price 1765). Also, although Smith and Price shared many mutual friends (e.g. David Hume) and moved in the same London social circles, Smith himself had a low opinion of Price. In a letter dated 22 December 1785, Smith writes: “Price’s speculations cannot fail to sink into the neglect that they have always deserved. I have always considered him as a factious citizen, a most superficial Philosopher and by no means an able calculator.” (Corr. No. 251) Ouch!

Nevertheless, although there is no other evidence that Smith engaged with or incorporated Bayesian probability theory in any of his works, he definitely had some implicit appreciation of the concept of probability, for the words “probability” and “probably” appear a lot of times in his Wealth of Nations — by my count, over 110 times in all! By way of comparison, the word “liberty” appears 88 times. (By way of further comparison, “probability” and “probably” appear about two dozen times in Smith’s Theory of Moral Sentiments, while the word “liberty” appears only 15 times.) Bonus link: Check out this paper by my colleague Michael Emmett Brady (2016), a lecturer at the Dominguez Hills campus of California State University.

Talk about the division of labor: large language models can write your term papers and grade them too!

Is American exceptionalism for real? With the recent news that the public debt of the United States has now surpassed $40 trillion (see here & here, for example), I want to take a moment to share my previous blog posts (from earlier this year) on Adam Smith’s thoughts on public debts:

1. “The last chapter of The Wealth of Nations: *Of Publick Debts*“
2. “Adam Smith, father of public choice theory“
3. “The real reason why nations fail according to Adam Smith“
4. “Adam Smith on the social cost of public debts“
5. “Adam Smith’s new Utopia (and his epic smack-down of politicians)“