prior probability

In my previous post, we saw Frank Knight’s critique of A. C. Pigou’s economic analysis of traffic congestion. For Pigou, road congestion is a tragedy of the commons or market failure that requires active government intervention: the imposition of a tax or congestion fee to reduce the number of drivers on the road. For Knight (and for his disciple, Coase), however, road congestion is a legal failure: the absence of property rights in roadways. Today, I want to take a closer look at the logic of Knight’s critique of Pigou.

To begin, why does Knight tweak Pigou’s hypothetical example of road congestion? In Pigou’s original setup (Pigou 1920, p. 194), “there are two roads ABD and ACD both leading from A to D.” Knight, by contrast, modifies Pigou’s hypothetical example. In Knight’s modified hypo, one of the roads is shoddy and full of potholes but wide enough to handle a lot of extra traffic, while the other road is in perfect condition but too narrow to accommodate too much traffic:

“Suppose between two points there are two highways, one of which is broad enough to accommodate without crowding all the traffic which may care to use it, but is poorly graded and surfaced, while the other is a much better road but narrow and quite limited in capacity.” (Knight 1924, p. 584; Knight 1952, p. 162; footnote omitted)

Next, Knight concedes that “the narrower and better road” will become congested because too many drivers (“too many”, that is, relative to the narrow road’s optimal carrying capacity) will prefer the good road over the bad one:

“As more trucks use the narrower and better road, congestion develops, until at a certain point it becomes equally profitable to use the broader but poorer highway. The congestion and interference resulting from the addition of any particular truck to the stream of traffic on the narrow but good road affects in the same way the cost and output of all the trucks using that road.” (Knight 1924, p. 585; Knight 1952, p. 162)

But then, after summarizing Pigou’s market failure argument and Pigou’s preferred solution (i.e. the imposition of a congestion fee by the government ), Knight explains why this road congestion example is just as much a legal failure (the absence of property rights) as it is a market failure (tragedy of the commons):

“The owner of a superior opportunity for investment can set the charge for its use at any amount not greater than the excess of the product of the first unit of investment above what that unit could produce on the free opportunity. Under this charge investment will flow into the superior road up to the point where congestion and diminishing returns set in.” (Knight 1924, p. 587; Knight 1952, p. 164)

Here comes the crucial masterstroke, a move that Coase must have taken close notice and made a careful note of at some point in his storied academic career. Knight reframes the traffic congestion example by turning it on its head:

By reducing the charge, the owner will increase the amount of traffic using his road (or in general the amount of investment of labor and capital in any opportunity). But obviously the owner of the road will not set the charge so low that the last truck which uses the road secures a return in excess of the amount which it adds to the total product of the road (that is, of all the trucks which use it). This is clearer if we think of the owner of the road hiring the trucks instead of their hiring the use of the road. The effect is the same either way; it is still the same if some third party hires the use of both. The toll or rent will be so adjusted that added product of the last truck which uses the narrow road is just equal to what it could produce on the broad road. No truck will pay a higher charge, and it is not to the interest of the owner of the road to accept a lower fee. And this adjustment is exactly that which maximizes the total product of both roads.

Although Knight does (yet!) not use the actual word “reciprocal”, the upside-down Alice-in-wonderland logic of his reasoning in this crucial passage is undeniable, for Knight is telling us that we can frame the traffic congestion problem in one of two ways: as drivers independently choosing which route to take in order to reach their destination, or as road owners choosing how many truck drivers to allow on their roads, and whether we adopt the point of view of the truck drivers or the point of view of the road owner, costs are going to be unavoidable!

In fact, it is at this crucial point in his devastating takedown of Pigou — that is, right after explaining the reciprocal logic of his reasoning — that Knight introduces three diagrams to the depict congestion costs in his hypothetical road example and uses the word “reciprocal” to describe the relationship between the cost curves in two of the diagrams. (For reference, the two relevant diagrams and cost curves are pictured below.)

Notice how the cost curves in Knight’s diagrams are mirror images of each other. Why are they reverse images of each other? Because one is expressed in terms of outputs while the other is expressed in terms of inputs: the upward-sloping curve in the chart on the left (Chart C) “is a cost curve for the narrow road, showing the cost of successive units of output” (Knight 1924, p. 588; Knight 1952, p. 165; emphasis in the original), while the downward-sloping curve in the chart on the right (Chart D) “represents the same facts as Chart C [the chart on the left] but in terms of the product of successive units of investment [i.e. inputs] instead of the cost of successive units of output.” (Knight 1924, p. 589; Knight 1952, p. 165)

In short, the cost curves in Knight’s diagrams depict the reciprocal relationship between road owners and truck drivers, i.e. the reciprocal logic of “the owner of the road hiring the trucks” and the truck drivers “hiring the use of the road.” More importantly, in the words of Frank Knight, words that must have resonated with Ronald Coase at some point in his academic career: “The argument is the same [i.e. whether we think of the owner of the road hiring the trucks instead of their hiring the use of the road], but stated in inverse or reciprocal form.” (Knight 1924, p. 589; Knight 1952, p. 166; my emphasis)

But even if Ronald Coase was influenced by Knight’s reasoning (after all, Coase and Knight’s papers have similar titles and attack a common enemy, Pigou), Coase goes a step further — one big step further — for Coase frames all harms as a reciprocal problem. Why and when did Coase make this leap? I will present my own conjecture regarding the origins of Coase’s reciprocal-harm axiom when I conclude this series in my next post.

Was it the University of Cambridge economist A. C. Pigou (pictured below) who unwittingly inspired Ronald Coase’s notion of reciprocal harms?

We spent most of last week tracing the origins of Ronald Coase’s axiom that harms are a reciprocal problem, i.e. the counter-intuitive idea that victims are just as responsible for harms as wrongdoers. Among other possibilities, potential precursors that may have inspired Coase include (1) Wesley Hohfeld’s theory of legal relations, (2) what economist Jim Buchanan has called the “LSE tradition in cost theory“, and (3) Sir Arnold Plant, the British economist who had the largest influence on Coase’s intellectual development.

Today, it’s Pigou’s turn, for Coase’s critique of Pigou in “The Problem of Social Cost” — i.e. the work that contains Coase’s reciprocal-harm axiom — can be traced back to a paper by Frank H. Knight titled “Some Fallacies in the Interpretation of Social Cost” (Knight 1924, 1952), which was first published in the Quarterly Journal of Economics in 1924 and reprinted in 1952 in a collection of essays Readings in Price Theory edited by George J. Stigler and Kenneth Boulding. To begin, as I explained in my previous post in this series, we now know that the title of Ronald Coase’s paper “The Problem of Social Cost” — i.e. the work that contains Coase’s reciprocal-harm axiom — can be traced back to Knight’s social cost paper.

But in addition to similar titles, both Knight’s paper and Coase’s also share a common intellectual enemy: the English economist A. C. Pigou. By way of background, Pigou had created an entirely new field of economics — now known as welfare economics — when he published his landmark work The Economics of Welfare in 1920. (Pigou’s textbook is available here. As an aside, the word “welfare” in this context refers to the general well-being of society overall, not to specific welfare programs like Social Security, farm subsidies, etc.) Pigou had used the example of traffic congestion in the first edition of his textbook to illustrate the general problem of market failure and to make the case for government intervention in order to correct such failures:

“Suppose there are two roads ABD and ACD both leading from A to D. If left to itself, traffic would be so distributed that the trouble involved in driving a ‘representative’ cart along each of the two roads would be equal. But, in some circumstances, it would be possible, by shifting a few carts from route B to route C, greatly to lessen the trouble of driving those still left on B, while only slightly increasing the trouble of driving along C. In these circumstances a rightly chosen measure of differential taxation against road B would create an ‘artificial’ situation superior to the ‘natural’ one. But the measure of differentiation must be rightly chosen.” (Pigou 1920, p. 194)

Pigou thus uses the example of road congestion in his welfare economics textbook to make two crucial points: (1) congestion is a market failure because each individual driver imposes time costs on other drivers, and (2) the government can solve this problem by imposing a corrective tax on the overcrowded road in order to divert traffic into the less crowded road. But both Coase and Knight in their respective social cost papers would launch blistering attacks against Pigou. Here is Knight’s opening salvo against Pigou:

“Professor Pigou’s logic in regard to the roads is, as logic, quite unexceptionable. Its weakness is one frequently met with in economic theorizing, namely that the assumptions diverge in essential respects from the facts of real economic situations. If the roads are assumed to be subject to private appropriation and exploitation, precisely the ideal situation which would be established by the imaginary tax will be brought about through the operation of ordinary economic motives The owner of the broad road could not under effective competition charge anything for its use. If an agency of production is not subject to diminishing returns, and cannot be monopolized, there is, in fact, no incentive to its appropriation, and it will remain a free good. But the owner of the narrow road can charge for its use a toll representing its ‘superiority’ over the free road, in accordance with the theory of rent, which is as old as Ricardian economics.” (Knight 1924, pp. 584-585; Knight 1952, pp. 162-163; footnote omitted; my emphases)

In other words, the problem of traffic congestion is not so much a market failure as a legal one: the lack of property rights over roads! If roads were privately owned, the owner of the narrow, congested road could reduce congestion and thus solve the externality by charging a toll — i.e. by imposing a cost on the drivers of the narrow road — without the need for any further government intervention (beyond the creation and enforcement of property rights in roads). For Knight, no government intervention at all is required to solve the so-called market failure in Pigou’s traffic congestion example.

Knight then illustrates his legal failure argument — as well as his devastating critique of Pigou’s market failure argument — with a diagram, and he will use the word “reciprocal” to describe the relationship between two of the cost curves in his diagram. I will reproduce Knight’s diagram and discuss their possible significance for Coase in my next post. (To be continued …)

Thus far this past week, we have attempted to trace the intellectual origins of Ronald Coase’s counter-intuitive idea of reciprocal harms to his undergraduate years at the London School of Economics (see here and here), but as I asked at the end of my previous post, what if the English economist had developed his model of reciprocal harms after his undergraduate years at the LSE? More specifically, another possible catalyst of Coase’s reciprocal-harm axiom is Chicago economist Frank H. Knight (pictured below), especially his paper “Some Fallacies in the Interpretation of Social Cost” (Knight 1924, 1952), which was first published in the Quarterly Journal of Economics in 1924.

Although it is unlikely that Coase would have read Knight’s 1924 paper as a student at LSE (after all, Coase’s major was not economics), Knight’s social cost paper was considered important enough to be subsequently included in a collection of essays edited by George J. Stigler & Kenneth Boulding and published by the American Economics Association in 1952 (PDF here), so even if Coase was not familiar with the original version of Knight’s paper he may still have become acquainted with it in its subsequent incarnation. In addition, consider the following exchange that took place between Ronald Coase and his Chicago colleague William Landes in 1981:

LANDES: Ronald, you mentioned Frank Knight’s Risk, Uncertainty, and Profit as an important influence on your work. One of Knight’s papers that has been most influential in the recent law and economics area is his paper, “Some Fallacies in the Interpretation of Social Cost.” That was in some ways a forerunner of your paper on social cost because it spells out the importance of property rights in resource allocation, and I wonder if you were familiar with Knight’s article?

COASE: Oh, yes. In fact, I would say that the title of my paper came from Frank Knight, and the title of the paper was rather to indicate the topic I was talking about, because, of course, I don’t think the concept of social cost is a very useful one, and I don’t ever refer to it. But it did indicate to people what I was talking about. I knew it, and if there are traces of what Knight says in my work, it wouldn’t surprise me.

(Source: page 183 of Edmund W. Kitch, “The Fire of Truth: A Remembrance of Law and Economics at Chicago, 1932-1970,” Journal of Law & Economics, Vol. 26, No. 1 (1983), pp. 163-234; PDF available here.)

The title of Ronald Coase’s social cost paper can thus be traced back to Frank Knight’s 1924 paper. But even more importantly, both social cost papers — Coase’s and Knight’s — share a common intellectual enemy: A. C. Pigou. I will explain how it was Knight’s devastating critique of Pigou that may have inspired Ronald Coase to frame harms as a “reciprocal” problem when I resume my series on Monday (5 October).

In my previous two posts (see here and here), we attempted to trace the intellectual origins of Ronald Coase’s reciprocal-harm axiom to Wesley Hohfeld’s theory of legal relations and to the “LSE tradition in cost theory” developed by Lionel Robbins and F. A. Hayek at the London School of Economics in the 1930s. A more likely source, however, is Arnold Plant (pictured below), Coase’s favorite professor at the London School of Economics.

Why Plant? For starters, because we have no evidence of Coase attending any lectures by Hayek or Robbins during his undergraduate years — in fact, Hayek did not join the faculty of the London School of Economics until 1931,[1] when Coase would have been studying abroad (see, e.g., Coase 1991)[2] — and furthermore, by his own admission (see, e.g., Coase 1982), it was Professor Plant who exerted the greatest influence on Coase’s intellectual development during his formative student years at the London School of Economics.[3]

By way of background, Coase had enrolled in the Bachelor of Commerce or “B.Com.” program at the London School of Economics in October 1929,[4] and Plant had begun teaching at the London School of Economics in 1930, when he was appointed to the newly‑created position as Sir Ernest Cassel Professor of Commerce with special reference to Business Administration.[5] (At the time, Plant’s teaching and research duties centered on the Industry and Trade Group in the B.Com.[6]) Here is where the paths of Ronald Coase and Arnold Plant intersect, for as part of his studies for the undergraduate B.Com. degree, the young Coase would have very likely enrolled in a course on the Elements of Commercial Law and Industrial Law,[7] a course that was most likely taught by Arnold Plant, who was the director of the Industry group at LSE at the time.[8]

As a result, Coase may have already been familiar with the case of the noisy confectioner (Sturges v. Bridgman) as well as with the many other reciprocal-harm cases in his 1960 social cost paper from his undergraduate student days at the London School of Economics, when Coase most likely attended a seminar on industrial law taught by Plant.[9] On this view, it was Plant who first taught Coase about the reciprocal nature of harms. But there is yet another possibility: what if Coase had developed his model of reciprocal harms after his undergraduate years at the LSE? (To be continued…)

[1] See Caldwell & Klausinger 2022, pp. 278-282.

[2] Coase had been awarded a travel scholarship for the 1931-32 academic year and was studying in the United States at that time.

[3] See Coase 1988a, p. 20. See also Marciano 2019, pp. 559-560.

[4] Coase had enrolled in an undergraduate commerce degree at the London School of Economics in 1929 and attended the lectures of Arnold Plant during his second year of studies. See Marciano 2019, p. 3.

[5] See Coase 1994, p. 179. See also Thomas 2016, p. 23.

[6] See Cord 2019, p. 332. In addition to the Industry Group, the B. Com. program at LSE comprised several other sections as well, including “Banking and Finance,” “General Transport,” “Shipping and Inland Transport,” “Public Utilities,” and “Art in Relation to Commerce.” See Appendix 3 in Thomas 2020.

[7] Coase himself once mentioned attending multiple law courses at the London School of Economics, where Coase studied commerce as an undergraduate from 1929 to 1931. See Coase 1988a, p. 29. See also Campbell & Klaes 2016, pp. 809–810. See also Marciano 2019, p. 557

[8] According to Jim Thomas (2020), an Emeritus Reader and Research Associate at the London School of Economics (LSE), the Bachelor of Commerce degree was a three‑year degree that was first offered by the LSE in 1919, and this undergraduate degree required three sets of exams: an Intermediate Examination at the end of the First Year as well as a Final Examination (the B.Com. Final) consisting of two parts, with Part 1 being taken at the end of the Second Year and Part 2 at the end of the Third Year. See Thomas 2020, pp. 6–7. Among the subjects that Coase would have been tested on for Part 1 of the B.Com Final was “Elements of Commercial Law (treated from the commercial rather than the legal standpoint).” See Thomas 2020, p. 19, Appendix 3(b). Likewise, for students in the Industry group, Part 2 of the B.Com Final Exam encompassed the following four subjects: (i) an “Approved Modern Foreign Language,” (ii) “Business Organization and Scientific Management,” (iii) “Works and Factory Accounting, with special reference to Cost Accounts and Depreciation,” and last but not least, (iv) “Industrial Law” or “The Law relating to Factories and Workshops, Workmen’s Compensation, Trade Unions, Employer’s Liability, Friendly Societies, National Insurance, [and] Labour conditions.” See Thomas 2020, pp. 20–21, Appendix 3(c). See also Campbell & Klaes 2016, pp. 809-810.

[9] See Marciano 2019, p. 557. See also Campbell & Klaes 2016, pp. 809-810.

Picking up where we last left off, I had identified another possible source of Ronald Coase’s counter-intuitive reciprocal-harm model at the end of my previous post: what economists and historians of economic thought call the “LSE tradition in cost theory,” an approach to economics that was developed by Lionel Robbins and F. A. Hayek at the London School of Economics in the 1930s. What is this LSE tradition, and how did it influence Coase’s own thinking?

In brief, both Robbins and Hayek had rejected the classical British view that cost is an objective or measurable physical quantity, such as labor hours or material units. Instead, they adopted a psychological, forward-looking “opportunity cost” approach to economics. Under this “opportunity cost” approach to economics, the true cost of any choice is not a historical tally of resources spent, but the subjective value of the next best alternative foregone at the moment of decision. (For further information, see the late Jim Buchanan’s overview “Introduction: L.S.E. Cost Theory in Retrospect,” in L.S.E. Essays on Costs. (Buchanan & Thirlby 1981, Ch. 1) This work is available here via the Online Library of Liberty (OLL) for free.)

On this view, one can thus trace a direct relationship between Robbins and Hayek’s opportunity cost approach to economics and Ronald Coase’s reciprocal model of harm: by applying the Robbins-Hayek-LSE view that cost is the value of the next best alternative foregone, Coase shows how preventing harm to one party always inflicts an opportunity cost on another! But is this what Coase learned during his student years at the London School of Economics, and if so, who did he learn it from, for as we shall see in my next post, we have no record of Coase ever attending any of Robbins or Hayek’s LSE lectures? (To be continued …)

The first time Ronald Coase uses the word “reciprocal” to describe the problem of harmful effects is in his 1959 FCC paper (Coase 1959), but this observation begs the question, what is the source of Coase’s reciprocal-harm model? Did Coase borrow or steal? Or did he discover it on his own? Today, I will one consider one possible source of Coase’s reciprocal-harm model: Wesley Hohfeld’s influential theory of legal relations. (Hohfeld 1913, 1917)

Among other things, Hohfeld (pictured below) observed the ubiquity of rights talk in legal reasoning as well as the polysemy of this legal concept: the legal concept of a “right” in law can be used in different ways to describe different types of legal relations. What Hohfeld did that would shape generations of legal scholars-and perhaps at least one economist, Coase–is create a theoretical framework to capture the precise meaning of different uses of the word “right” in legal reasoning.

Without getting bogged down in the tedious details of Wesley Hohfeld’s overall framework, the part of Hohfeld’s theory of legal relations that is most relevant to Coase’s reciprocal harm model is the logical relationship between legal rights and legal duties. Simply put, a Hohfeldian legal right is always linked to a correlative duty: a right held by one person, let’s call him A, necessarily entails the imposition of a corresponding legal duty on someone else, B.

Nevertheless, the available evidence indicates that Coase most likely developed his reciprocal harm model independent of the work Wesley Hohfeld. For starters, Coase never cites Hohfeld in either his FCC or social cost paper. (As an aside, the first British legal scholar to formally cite and integrate Hohfeld into English legal scholarship was G. W. Keeton (1930).)

Secondly, although Coase uses the word “right” dozens of times in both his FCC and social cost papers, for Coase a legal right is just another “factor of production” or economic input, such as land, labor, and capital. Towards the end of his social cost paper, Coase writes:

“If factors of production are thought of as rights. it becomes easier to understand that the right to do something which has a harmful effect (such as the creation of smoke, noise, smells, etc.) is also a factor of production. Just as we may use a piece of land in such a way as to prevent someone else from crossing it, or parking his car, or building his house upon it, so we may use it in such a way as to deny him a view or quiet or unpolluted air. The cost of exercising a right (of using a factor of production) is always the loss which is suffered elsewhere in consequence of the exercise of that right–the inability to cross land, to park a car, to build a house, to enjoy a view, to have peace and quiet or to breathe clean air. (Coase 1960, p. 44)

This passage, however, points us to another possible source of Coase’s reciprocal harm model: the so-called “LSE tradition in cost theory,” which was developed by Lionel Robbins and F. A. Hayek at the London School of Economics in the 1930s. (To be continued…)

Works cited:

R. H. Coase. “The Federal Communications Commission.” Journal of Law and Economics, Vol. 2 (1959), pp. 1-41.

R. H. Coase. “The Problem of Social Cost.” Journal of Law & Economics, Vol. 3 (1960), pp. 1-44.

W. N. Hohfeld. “Some Fundamental Legal Conceptions as Applied in Judicial Reasoning.” Yale Law Journal, Vol. 23, No. 1 (1913), pp. 16-59.

W. N. Hohfeld. “Fundamental Legal Conceptions as Applied in Judicial Reasoning.” Yale Law Journal, Vol. 26, No. 8 (1917), pp. 710-770.

G. W. Keeton. 1930. The Elementary Principles of Jurisprudence. London: A. C. Black.

I have long been fascinated by the English economist Ronald Coase’s insight that harms are always a “reciprocal” or jointly-caused problem–so much so that I wrote a paper at the end of last year titled “Coase’s fable” in which I describe the idea of reciprocal harms as an axiom or truism, trace the intellectual origins of Coase’s counter-intuitive idea, and explore its domain or outer limits. (For reference, I blogged about my 2025 Coase paper here, here, here, and here. Also, for the record, I have not used any A.I. models to research or write any of my papers!)

Since then, I presented my paper in May of 2026 at the first-ever joint conference of the European Society for the History of Economic Thought (ESHET) and the History of Economics Society (HES) at the Université Côte d’Azur in Nice, France, and based on the excellent feedback I received at the conference, I have expanded the part of my paper that traces the intellectual origins of Coase’s reciprocal-harm model. (Among other things, I explore the relationship between Coase’s model and the opportunity-cost approach to economics that Lionel Robbins and F. A. Hayek developed at the London School of Economics in the 1930s, and I consider the possibility that Coase might have been influenced by Wesley Hohfeld’s influential theory of legal relations — stated simply, the idea that every legal right creates a corresponding duty, and vice versa, every legal duty implies a corresponding right.) I will say more about these substantive additions to my paper in my next two posts; in the meantime, here is a link my revised paper (via SSRN).

On this day (28 September) in 1821, Juan José Espinosa de los Monteros, secretary of Mexico’s newly-created “Provisional Governmental Board” (the first governing body of independent Mexico that administered the country from 28 September 1821 until 24 February 1822), drafts the Acta de Independencia del Imperio Mexicano (Declaration of Independence of the Mexican Empire; pictured below), the founding document by which Mexico declared independence from the Spanish Empire.

Before there was Bad Bunny, there was Tego Calderón!