Category Archives: History of Economic Thought

On Gary Becker

Gary Becker, as you must surely know by now, has passed away. This is an incredible string of bad luck for the University of Chicago. With Coase and Fogel having passed recently, and Director, Stigler and Friedman dying a number of years ago, perhaps Lucas and Heckman are the only remaining giants from Chicago’s Golden Age.

Becker is of course known for using economic methods – by which I mean constrained rational choice – to expand economics beyond questions of pure wealth and prices to question of interest to social science at large. But this contribution is too broad, and he was certainly not the only one pushing such an expansion; the Chicago Law School clearly was doing the same. For an economist, Becker’s principal contribution can be summarized very simply: individuals and households are producers as well as consumers, and rational decisions in production are as interesting to analyze as rational decisions in consumption. As firms must purchase capital to realize their productive potential, humans much purchase human capital to improve their own possible utilities. As firms take actions today which alter constraints tomorrow, so do humans. These may seem to be trite statements, but that are absolutely not: human capital, and dynamic optimization of fixed preferences, offer a radical framework for understanding everything from topics close to Becker’s heart, like educational differences across cultures or the nature of addiction, to the great questions of economics like how the world was able to break free from the dreadful Malthusian constraint.

Today, the fact that labor can augment itself with education is taken for granted, which is a huge shift in how economists think about production. Becker, in his Nobel Prize speech: “Human capital is so uncontroversial nowadays that it may be difficult to appreciate the hostility in the 1950s and 1960s toward the approach that went with the term. The very concept of human capital was alleged to be demeaning because it treated people as machines. To approach schooling as an investment rather than a cultural experience was considered unfeeling and extremely narrow. As a result, I hesitated a long time before deciding to call my book Human Capital, and hedged the risk by using a long subtitle. Only gradually did economists, let alone others, accept the concept of human capital as a valuable tool in the analysis of various economic and social issues.”

What do we gain by considering the problem of human capital investment within the household? A huge amount! By using human capital along with economic concepts like “equilibrium” and “private information about types”, we can answer questions like the following. Does racial discrimination wholly reflect differences in tastes? (No – because of statistical discrimination, underinvestment in human capital by groups that suffer discrimination can be self-fulfilling, and, as in Becker’s original discrimination work, different types of industrial organization magnify or ameliorate tastes for discrimination in different ways.) Is the difference between men and women in traditional labor roles a biological matter? (Not necessarily – with gains to specialization, even very small biological differences can generate very large behavioral differences.) What explains many of the strange features of labor markets, such as jobs with long tenure, firm boundaries, etc.? (Firm-specific human capital requires investment, and following that investment there can be scope for hold-up in a world without complete contracts.) The parenthetical explanations in this paragraph require completely different policy responses from previous, more naive explanations of the phenomena at play.

Personally, I find human capital most interesting in understanding the Malthusian world. Malthus conjectured the following: as productivity improved for some reason, excess food will appear. With excess food, people will have more children and population will grow, necessitating even more food. To generate more food, people will begin farming marginal land, until we wind up with precisely the living standards per capita that prevailed before the productivity improvement. We know, by looking out our windows, that the world in 2014 has broken free from Malthus’ dire calculus. But how? The critical factors must be that as productivity improves, population does not grow, or else grows slower than the continued endogenous increases in productivity. Why might that be? The quantity-quality tradeoff. A productivity improvement generates surplus, leading to demand for non-agricultural goods. Increased human capital generates more productivity on those goods. Parents have fewer kids but invest more heavily in their human capital so that they can work in the new sector. Such substitution is only partial, so in order to get wealthy, we need a big initial productivity improvement to generate demand for the goods in the new sector. And thus Malthus is defeated by knowledge.

Finally, a brief word on the origin of human capital. The idea that people take deliberate and costly actions to improve their productivity, and that formal study of this object may be useful, is modern: Mincer and Schultz in the 1950s, and then Becker with his 1962 article and famous 1964 book. That said, economists (to the chagrin of some other social scientists!) have treated humans as a type of capital for much longer. A fascinating 1966 JPE [gated] traces this early history. Petty, Smith, Senior, Mill, von Thunen: they all thought an accounting of national wealth required accounting for the productive value of the people within the nation, and 19th century economists frequently mention that parents invest in their children. These early economists made such claims knowing they were controversial; Walras clarifies that in pure theory “it is proper to abstract completely from considerations of justice and practical expediency” and to regard human beings “exclusively from the point of view of value in exchange.” That is, don’t think we are imagining humans as being nothing other than machines for production; rather, human capital is just a useful concept when discussing topics like national wealth. Becker, unlike the caricature where he is the arch-neoliberal, was absolutely not the first to “dehumanize” people by rationalizing decisions like marriage or education in a cost-benefit framework; rather, he is great because he was the first to show how powerful an analytical concept such dehumanization could be!

“X-Efficiency,” M. Perelman (2011)

Do people still read Leibenstein’s fascinating 1966 article “Allocative Efficiency vs. X-Efficiency”? They certainly did at one time: Perelman notes that in the 1970s, this article was the third-most cited paper in all of the social sciences! Leibenstein essentially made two points. First, as Harberger had previously shown, distortions like monopoly simply as a matter of mathematics can’t have large welfare impacts. Take monopoly. for instance. The deadweight loss is simply the change in price times the change in quantity supplied times .5 times the percentage of the economy run by monopolist firms. Under reasonable looking demand curves, those deadweight triangles are rarely going to be even ten percent of the total social welfare created in a given industry. If, say, twenty percent of the final goods economy is run by monopolists, then, we only get a two percent change in welfare (and this can be extended to intermediate goods with little empirical change in the final result). Why, then, worry about monopoly?

The reason to worry is Leibenstein’s second point: firms in the same industry often have enormous differences in productivity, and there is tons of empirical evidence that firms do a better job of minimizing costs when under the selection pressures of competition (Schmitz’ 2005 JPE on iron ore producers provides a fantastic demonstration of this). Hence, “X-inefficiency”, which Perelman notes is named after Tolstoy’s “X-factor” in the performance of armies from War and Peace, and not just just allocative efficiency may be important. Draw a simple supply-demand graph and you will immediately see that big “X-inefficiency rectangles” can swamp little Harberger deadweight loss triangles in their welfare implications. So far, so good. These claims, however, turned out to be incredibly controversial.

The problem is that just claiming waste is really a broad attack on a fundamental premise of economics, profit maximization. Stigler, in his well-named X-istence of X-efficiency (gated pdf), argues that we need to be really careful here. Essentially, he is suggesting that information differences, principal-agent contracting problems, and many other factors can explain dispersion in costs, and that we ought focus on those factors before blaming some nebulous concept called waste. And of course he’s correct. But this immediately suggests a shift from traditional price theory to a mechanism design based view of competition, where manager and worker incentives interact with market structure to produce outcomes. I would suggest that this project is still incomplete, that the firm is still too much of a black box in our basic models, and that this leads to a lot of misleading intuition.

For instance, most economists will agree that perfectly price discriminating monopolists have the same welfare impact as perfect competition. But this intuition is solely based on black box firms without any investigation of how those two market structures affect the incentive for managers to collect costly information of efficiency improvements, on the optimal labor contracts under the two scenarios, etc. “Laziness” of workers is an equilibrium outcome of worker contracts, management monitoring, and worker disutility of effort. Just calling that “waste” as Leibenstein does is not terribly effective analysis. It strikes me, though, that Leibenstein is correct when he implicitly suggests that selection in the marketplace is more primitive than profit maximization: I don’t need to know much about how manager and worker incentives work to understand that more competition means inefficient firms are more likely to go out of business. Even in perfect competition, we need to be careful about assuming that selection automatically selects away bad firms: it is not at all obvious that the efficient firms can expand efficiently to steal business from the less efficient, as Chad Syverson has rigorously discussed.

So I’m with Perelman. Yes, Leibenstein’s evidence for X-inefficiency was weak, and yes, he conflates many constraints with pure waste. But on the basic points – that minimized costs depend on the interaction of incentives with market structure instead of simply on technology, and that heterogeneity in measured firm productivity is critical to economic analysis – Leibenstein is far more convincing that his critics. And while Syverson, Bloom, Griffith, van Reenen and many others are opening up the firm empirically to investigate the issues Leibenstein raised, there is still great scope for us theorists to more carefully integrate price theory and mechanism problems.

Final article in JEP 2011 (RePEc IDEAS). As always, a big thumbs up to the JEP for making all of their articles ungated and free to read.

On Coase’s Two Famous Theorems

Sad news today that Ronald Coase has passed away; he was still working, often on the Chinese economy, at the incredible age of 102. Coase is best known to economists for two statements: that transaction costs explain many puzzles in the organization of society, and that pricing for durable goods presents a particular worry since even a monopolist selling a durable good needs to “compete” with its future and past selves. Both of these statements are horribly, horribly misunderstood, particularly the first.

Let’s talk first about transaction costs, as in “The Nature of the Firm” and “The Problem of Social Cost”, which are to my knowledge the most cited and the second most cited papers in economics. The Problem of Social Cost leads with its famous cattle versus crops example. A farmer wishes to grow crops, and a rancher wishes his cattle to roam where the crops grow. Should we make the rancher liable for damage to the crops (or restrain the rancher from letting his cattle roam at all!), or indeed ought we restrain the farmer from building a fence where the cattle wish to roam? Coase points out that in some sense both parties are causally responsible for the externality, that there is some socially efficient amount of cattle grazing and crop planting, and that if a bargain can be reached costlessly, then there is some set of side payments where the rancher and the farmer are both better off than having the crops eaten or the cattle fenced. Further, it doesn’t matter whether you give grazing rights to the cattle and force the farmer to pay for the “right” to fence and grow crops, or whether you give farming rights and force the rancher to pay for the right to roam his cattle.

This basic principle applies widely in law, where Coase had his largest impact. He cites a case where confectioner machines shake a doctor’s office, making it impossible for the doctor to perform certain examinations. The court restricts the ability of the confectioner to use the machine. But Coase points out that if the value of the machine to the confectioner exceeds the harm of shaking to the doctor, then there is scope for a mutually beneficial side payment whereby the machine is used (at some level) and one or the other is compensated. A very powerful idea indeed.

Powerful, but widely misunderstood. I deliberately did not mention property rights above. Coase is often misunderstood (and, to be fair, he does at points in the essay imply this misunderstanding) as saying that property rights are important, because once we have property rights, we have something that can “be priced” when bargaining. Hence property rights + externalities + no transaction costs should lead to no inefficiency if side payments can be made. Dan Usher famously argued that this is “either tautological, incoherent, or wrong”. Costless bargaining is efficient tautologically; if I assume people can agree on socially efficient bargains, then of course they will. The fact that side payments can be agreed upon is true even when there are no property rights at all. Coase says that “[i]t is necessary to know whether the damaging business is liable or not for damage since without the establishment of this initial delimitation of rights there can be no market transactions to transfer and recombine them.” Usher is correct: that statement is wrong. In the absence of property rights, a bargain establishes a contract between parties with novel rights that needn’t exist ex-ante.

But all is not lost for Coase. Because the real point of his paper begins with Section VI, not before, when he notes that the case without transaction costs isn’t the interesting one. The interesting case is when transaction costs make bargaining difficult. What you should take from Coase is that social efficiency can be enhanced by institutions (including the firm!) which allow socially efficient bargains to be reached by removing restrictive transaction costs, and particularly that the assignment of property rights to different parties can either help or hinder those institutions. One more thing to keep in mind about the Coase Theorem (which Samuelson famously argued was not a theorem at all…): Coase implicitly is referring to Pareto efficiency in his theorem, but since property rights are an endowment, we know from the Welfare Theorems that benefits exceeds costs is not sufficient for maximizing social welfare.

Let’s now consider the Coase Conjecture: this conjecture comes, I believe, from a very short 1972 paper, Durability and Monopoly. The idea is simple and clever. Let a monopolist own all of the land in the US. If there was a competitive market in land, the price per unit would be P and all Q units will be sold. Surely a monopolist will sell a reduced quantity Q2 less than Q at price P2 greater than P? But once those are sold, we are in trouble, since the monopolist still has Q-Q2 units of land. Unless the monopolist can commit to never sell that additional land, we all realize he will try to sell it sometime later, at a new maximizing price P3 which is greater than P but less than P2. He then still has some land left over, which he will sell even cheaper in the next period. Hence, why should anyone buy in the first period, knowing the price will fall (and note that the seller who discounts the future has the incentive to make the length between periods of price cutting arbitrarily short)? The monopolist with a durable good is thus unable to make rents. Now, Coase essentially never uses mathematical theorems in his papers, and you game theorists surely can see that there are many auxiliary assumptions about beliefs and the like running in the background here.

Luckily, given the importance of this conjecture to pricing strategies, antitrust, auctions, etc., there has been a ton of work on the problem since 1972. Nancy Stokey (article gated) has a famous paper written here at MEDS showing that the conjecture only holds strictly when the seller is capable of selling in continuous time and the buyers are updating beliefs continuously, though approximate versions of the conjecture hold when periods are discrete. Gul, Sonnenschein and Wilson flesh out the model more completely, generally showing the conjecture to hold in well-defined stationary equilibrium across various assumptions about the demand curve. McAfee and Wiseman show in a recent ReStud that even the tiniest amount of “capacity cost”, or a fee that must be paid in any period for X amount of capacity (i.e., the need to hire sales agents for the land), destroys the Coase reasoning. The idea is that in the final few periods, when I am selling to very few people, even a small capacity cost is large relative to the size of the market, so I won’t pay it; backward inducting, then, agents in previous periods know it is not necessarily worthwhile to wait, and hence they buy earlier at the higher price. It goes without saying that there are many more papers in the formal literature.

(Some final notes: Coase’s Nobel lecture is well worth reading, as it summarizes the most important thread in his work: “there [are] costs of using the pricing mechanism.” It is these costs that explain why, though markets in general have such amazing features, even in capitalist countries there are large firms run internally as something resembling a command state. McCloskey has a nice brief article which generally blames Stigler for the misunderstanding of Coase’s work. Also, while gathering some PDFs for this article, I was shocked to see that Ithaka, who run JSTOR, is now filing DMCA takedowns with Google against people who host some of these legendary papers (like “Problem of Social Cost”) on their academic websites. What ridiculousness from a non-profit that claims its mission is to “help the academic community use digital technologies to preserve the scholarly record.”)

Paul Samuelson’s Contributions to Welfare Economics, K. Arrow (1983)

I happened to come across a copy of a book entitled “Paul Samuelson and Modern Economic Theory” when browsing the library stacks recently. Clear evidence of his incredible breadth are in the section titles: Arrow writes about his work on social welfare, Houthhaker on consumption theory, Patinkin on money, Tobin on fiscal policy, Merton on financial economics, and so on. Arrow’s chapter on welfare economics was particularly interesting. This book comes from the early 80s, which is roughly the end of social welfare as a major field of study in economics. I was never totally clear on the reason for this – is it simply that Arrow’s Possibility Theorem, Sen’s Liberal Paradox, and the Gibbard-Satterthwaite Theorem were so devastating to any hope of “general” social choice rules?

In any case, social welfare is today little studied, but Arrow mentions a number of interesting results which really ought be better known. Bergson-Samuelson, conceived when the two were in graduate school together, is rightfully famous. After a long interlude of confused utilitarianism, Pareto had us all convinced that we should dismiss cardinal utility and interpersonal utility comparisons. This seems to suggest that all we can say about social welfare is that we should select a Pareto-optimal state. Bergson and Samuelson were unhappy with this – we suggest individuals should have preferences which represent an order (complete and transitive) over states, and the old utilitarians had a rule which imposed a real number for society’s value of any state (hence an order). Being able to order states from a social point of view seems necessary if we are to make decisions. Some attempts to extend Pareto did not give us an order. (Why is an order important? Arrow does not discuss this, but consider earlier attempts at extending Pareto like Kaldor-Hicks efficiency: going from state s to state s’ is KH-efficient if there exist ex-post transfers under which the change is Paretian. Let person a value the bundle (1,1)>(2,0)>(1,0)>all else, and person b value the bundle (1,1)>(0,2)>(0,1)>all else. In state s, person a is allocated (2,0) and person b (0,1). In state s’, person a is allocated (1,0) and person b is allocated (0,2). Note that going from s to s’ is a Kaldor-Hicks improvement, but going from s’ to s is also a Kaldor-Hicks improvement!)

Bergson and Samuelson wanted to respect individual preferences – society can’t prefer s to s’ if s’ is a Pareto improvement on s in the individual preference relations. Take the relation RU. We will say that sRUs’ if all individuals weakly prefer s to s’. Not that though RU is not complete, it is transitive. Here’s the great, and non-obvious, trick. The Polish mathematician Szpilrajn has a great 1930 theorem which says that if R is a transitive relation, then there exists a complete relation R2 which extends R; that is, if sRs’ then sR2s’, plus we complete the relation by adding some more elements. This is not a terribly easy proof, it turns out. That is, there exists social welfare orders which are entirely ordinal and which respect Pareto dominance. Of course, there may be lots of them, and which you pick is a problem of philosophy more than economics, but they exist nonetheless. Note why Arrow’s theorem doesn’t apply: we are starting with given sets of preferences and constructing a social preference, rather than attempting to find a rule that maps any individual preferences into a social rule. There have been many papers arguing that this difference doesn’t matter, so all I can say is that Arrow himself, in this very essay, accepts that difference completely. (One more sidenote here: if you wish to start with individual utility functions, we can still do everything in an ordinal way. It is not obvious that every indifference map can be mapped to a utility function, and not even true without some type of continuity assumption, especially if we want the utility functions to themselves be continuous. A nice proof of how we can do so using a trick from probability theory is in Neuefeind’s 1972 paper, which was followed up in more generality by Mount and Reiter here at MEDS then by Chichilnisky in a series of papers. Now just sum up these mapped individual utilities, and I have a Paretian social utility function which was constructed entirely in an ordinal fashion.)

Now, this Bergson-Samuelson seems pretty unusable. What do we learn that we don’t know from a naive Pareto property? Here are two great insights. First, choose any social welfare function from the set we have constructed above. Let individuals have non-identical utility functions. In general, there is no social welfare function which is maximized by always keeping every individual’s income identical in all states of the world! The proof of this is very easy if we use Harsanyi’s extension of Bergson-Samuelson: if agents are Expected Utility maximizers, than any B-S social welfare function can be written as the weighted linear combination of individual utility functions. As relative prices or the social production possibilities frontier changes, the weights are constant, but the individual marginal utilities are (generically) not. Hence if it was socially optimal to endow everybody with equal income before the relative price change, it (generically) is not later, no matter which Pareto-respecting measure of social welfare your society chooses to use! That is, I think, an astounding result for naive egalitarianism.

Here’s a second one. Surely any good economist knows policies should be evaluated according to cost-benefit analysis. If, for instance, the summed willingness-to-pay for a public good exceeds the cost of the public good, then society should buy it. When, however, does a B-S social welfare function allow us to make such an inference? Generically, such an inference is only possible if the distribution of income is itself socially optimal, since willingness-to-pay depends on the individual budget constraints. Indeed, even if demand estimation or survey evidence suggests that there is very little willingness-to-pay for a public good, society may wish to purchase the good. This is true even if the underlying basis for choosing the particular social welfare function we use has nothing at all to do with equity, and further since the B-S social welfare function respects individual preferences via the Paretian criterion, the reason we build the public good also has nothing to do with paternalism. Results of this type are just absolutely fundamental to policy analysis, and are not at all made irrelevant by the impossibility results which followed Arrow’s theorem.

This is a book chapter, so I’m afraid I don’t have an online version. The book is here. Arrow is amazingly still publishing at the age of 91; he had an interesting article with the underrated Partha Dasgupta in the EJ a couple years back. People claim that relative consumption a la Veblen matters in surveys. Yet it is hard to find such effects in the data. Why is this? Assume I wish to keep up with the Joneses when I move to a richer place. If I increase consumption today, I am decreasing savings, which decreases consumption even more tomorrow. How my desire to change consumption today if I have richer peers then depends on that dynamic tradeoff, which Arrow and Dasgupta completely characterize.

“Innovation: The History of a Category,” B. Godin (2008)

What is innovation? What, indeed, is invention? I am confident that the average economist could not answer these questions. Is invention merely a novel process or idea? A novel process or idea for a given person? A new way of combining real resources like capital and labor? A new process which allows more of something to be created using a given amount of real resources? Does the new process need to be used, or embodied in technology, or is the idea enough?

None of these definitions seem satisfactory. A poem is a “new idea”, but we wouldn’t call it an invention. Novelty for a given person without technological embodiment, as a definition, doesn’t seem to distinguish between diffusion and simple learning. The idea of technology as a Solow residual means that merely using different mixtures of capital and labor to make the same product doesn’t qualify, and further the Solow residual includes things like Bowles-style adaptations to a more cooperative or trusting culture, which we generally don’t think of as innovation. Was Schumpeter correct that invention is a mere act of creativity “without importance to economic analysis”, or does the sequential nature of ideas mean that even non-embodied ideas are economically important?

In an interesting “genealogy of an idea”, Benoit Godin examines the history of how the terms invention and innovation were used in the Western World. The term invention goes back to Cicero, who listed the development of new argumentative concepts as one of the five tools of rhetoric. From the 15th to 19th centuries, invention was used occasionally to mean novel thoughts, but also novel recombinations (as in painting) or simple imitation (such as the patents given to importers in 18th century England).

It is really quite late in the game – well into the twentieth century – that something like “innovation is the invention, embodiment and diffusion of a commercial product” begins to be accepted as a definition. Part of this involves the shift from the individual inventor, the lone genius, to commercial firm R&D, as well as a recognition that simultaneous discovery and ex-post construction of credit meant that the lone genius inventor probably never existed. The terms discovery and invention began to separate. Science policy began to focus much more on the quantifiable, inventions as discoveries embodied in products or countable as patents. The word innovation became identified with an economic sense rather than an artistic sense which it previously possessed.’

Even the economic definition that would eventually be adopted is not the only one that could have developed. Schumpeter is often recognized as the father of economic studies of technological change, but his definition of innovation includes many concepts no longer covered by that term. For Schumpeter, innovation was tightly linked to creative destruction, or the dynamic ability of economic change to remake the commercial sphere. The opening of new commercial markets, for example, was an important part of innovation, whereas pure science was not.

http://www.csiic.ca/PDF/IntellectualNo1.pdf (2008 Working Paper – this is still unpublished, as far as I can tell).

A Note: The Paul Samuelson Papers

I’m currently visiting the Center for the History of Political Economy at Duke University. Duke’s library has been collecting the papers of famous economists for a number of years now, and the jewel of the collection, surely, are the Paul Samuelson papers, which I’m told are, by a great margin, the most popular collection in the entire Rare Books and Manuscripts department. I’ve been spending my lunches flipping through Paul’s work – despite 300 published papers, he nonetheless has many more fascinating unpublished papers, including a digression of various labor theories of value that really ought to be published somewhere given the frequent confusion of economists, both mainstream and Marxist, about that concept.

But most interesting are the correspondence, of which Samuelson appears to keep everything. Here are some of the more interesting lines.

On Joan Robinson and Marxists: Joan Robinson once said that the problem with Marxists is that, when you ask them whether constant capital is meant to be a stock or a flow, they respond, “Isn’t Marx a genius?”

On contrition: “You are right; I misspoke; I owe you a drink; I am ashamed.” This lovely triple-semicoloned sentence is not the result of, say, costing a friend a job, but rather a note to Uzawa about a minor technicality in the proof of a turnpike theorem!

On (weakly) efficient markets: “In sum, I am prone to believe in irrationality. But the organized markets do not give me much to bear out this belief. It is harder to play Keynes’ game than to look for fundamentals. Just try to guess which beauty contestant the others will find beautiful; often, the best you can do is look for those you think to be actually prettiest.”

On the role of economic proofs, in a letter to Samuelson from Arrow: “Necessary and sufficient conditions are not always useful.”

We also learn in a letter than Samuelson was told by all of his Chicago professors to go to Columbia, which though dominated by institutionalists, was a better home for a PhD student than Schumpeter’s Harvard. Perhaps we can apply Acemoglu’s Reversals of Fortune to economics departments: in the 1930s, Cambridge was on top, and Columbia was the graduate school of Arrow, Friedman, and almost Samuelson (though Friedman and, especially, Arrow spent much of their time either trying to get the department to offer more theory, or else taking classes in the mathematics department with Harold Hotelling!

If you are down in Carolina, it is well worth checking out Samuelson’s archives. It appears they only need a day’s notice to dig any particular set of documents up for you.

“Das Unsicherheitsmoment in der Wirtlehre,” K. Menger (1934)

Every economist surely knows the St. Petersburg Paradox described by Daniel Bernoulli in 1738 in a paper which can fairly claim to be the first piece of theoretical economics. Consider a casino offering a game of sequential coinflips that pays 2^(n-1) as a payoff if the first heads arrives on the nth flip of the coin. That is, if there is a heads on the first flip, you receive 1. If there is a tails on the first flip, and a heads on the second, you receive 2, and 4 if TTH, and 8 if TTTH, and so on. It is quite immediate that this game has expected payoff of infinity. Yet, Bernoulli points out, no one would pay anywhere near infinity for such a game. Why not? Perhaps they have what we would now call logarithmic utility, in which case I value the gamble at .5*ln(1)+.25*ln(2)+.125*ln(4)+…, a finite sum.

Now, here’s the interesting bit. Karl Menger proved in the 1927 that the standard response to the St. Petersburg paradox is insufficient (note that Karl with a K is the mathematically inclined son and mentor to Morganstern, rather than the relatively qualitative father, Carl, who somewhat undeservingly joined Walras and Jevons on the Mt. Rushmore of Marginal Utility). For instance, if the casino pays out e^(2^n-1) rather than 2^(n-1), then even an agent with logarithmic utility have infinite expected utility from such a gamble. This, nearly 200 years after Bernoulli’s original paper! Indeed, such a construction is possible for any unbounded utility function; let the casino pay out U^-1(2^(n-1)) when the first heads arrives on the nth flip, where U^-1 is inverse utility.

Things are worse, Menger points out. One can construct a thought experiment where, for any finite amount C and an arbitrarily small probability p, there is a bounded utility function where an agent will prefer the gamble to win some finite amount D with probability p to getting a sure thing of C [Sentence edited as suggested in the comments.] So bounding the utility function does not kill off all paradoxes of this type.

The 1927 lecture and its response are discussed in length in Rob Leonard’s “Von Neumann, Morganstern, and the Creation of Game Theory.” Apparently, Oskar Morganstern was at the Vienna Kreis where Menger first presented this result, and was quite taken with it, a fact surely interesting given Morganstern’s later development of expected utility theory. Indeed, one of Machina’s stated aims in his famous paper on EU with the Independence Axiom is providing a way around Menger’s result while salvaging EU analysis. If you are unfamiliar with Machina’s paper, one of the most cited in decision theory in the past 30 years, it may be worthwhile to read the New School HET description of the “fanning out” hypothesis which relates Machina to vN-M expected utility.

http://www.springerlink.com/content/m7q803520757q700/fulltext.pdf (Unfortunately, the paper above is both gated, and in German, as the original publication was in the formerly-famous journal Zeitschrift fur Nationalokonomie. The first English translation is in Shubik’s festschrift for Morganstern published in 1967, but I don’t see any online availability.)

“Sociology and Game Theory: Contemporary and Historical Perspectives,” R. Swedberg (2001)

Game theory is clearly the dominant theoretical paradigm in economics, and hugely important in political science as well, but does not appear to have made much of an impression on sociology. There is something odd about this. Swedberg quotes Max Weber, father of both our traditions, in an essay written just after 1900:

“The historical agent, to the extent that he is acting, as we are here assuming, in a strictly “rational” way, takes into account those “conditions” of the future course of events which interest him, which are “external” to him and, as far as he knows, given in reality. He then, in his mind, fits into the causal nexus various “possible” courses of action for himself, together with the consequences to be anticipated from them in combination with those “external” conditions, in order to decide on one or another of the courses of action appropriate to his “goal” in accordance with the “possible” outcomes which he has worked out in his mind.”

Weber here gives a very nice definition of the type of methodogically individualist decision theory which we economists use today. Yet 100 years later, game theory is not accepted in the mainstream of sociology. Why?

Richard Swedberg, in an essay for Theory and Society, attempts to answer that question. Tom Schelling, he of the social dilemma and the tipping point, unsurprisingly gained some traction with sociologists in the 1950s and 1960s. Both von Neumann and Morganstern, as well as Luce and Raiffa, were reviewed in the American Sociological Review, though the reviewer in both cases was an economist. Only a handful of models caught on, however. From my reading of Swedberg’s analysis, it seems four reasons were predominant.

First, the mathematical ability of sociologists (in that era, and to a lesser extent now) was too low to follow frontier developments in game theory. Second, many schools of sociology were, and are, uncomfortable with using such an individualist theory to analyze the outcome of social forces. Third, the use of game theory to make empirical predictions was less than successful – as Swedberg notes, game theory is better at explaning “why” than “how much?” Such connection with empirical reality was, and is, considered far more important by sociologists than economists. And fourth, some sociologists felt the axiomatics of game theory were far too demanding, and rather than expand the theory to incorporate incomplete information, or limits on reasoning power, or other factors that economists later studies, sociology simply left the technique behind. Swedberg argues that game theory in sociology is nonetheless a powerful tool for examining counterfactuals (something difficult to examine empirically, of course), and further that sociologists can contribute to game theory by discussing what the action spaces and payoffs look like in certain games, the ex-ante step that many economists do not take seriously enough. I would love to see precisely this sort of work by sociologists.

What is not made clear in the article, and perhaps is not well known to mathematical sociologists, are certain aspects of the history of game theory within economics. Sociologists were by no means alone in recognizing two big flaws in early game theory: the exact specification of the game form, including action spaces and payoffs, is very important, and the multiplicity of equilibria (or a nonunique core in cooperative games) makes analysis difficult. These two problems were to a large extent solved between the late 1970s and the late 1980s. The problem of specifying the game form in all its institutional detail was handled through the field of mechanism design, which roughly asks whether a planner or a principal can implement some outcome as the equilibrium of some game within a very large class. The robustness and refinement of equilibria led to a lot of work which helped us better understand what the Nash fixed point imposes in terms of decisionmaking, but ultimately led to something of a negative result: we will never have an equilibrium concept which is suitable for all types of games, and certainly not one which provides unique solutions.

Mechanism design and the robustness literature are interesting for two reasons. First, more or less 100 percent of the work in those areas was done by economists rather than by other social scientists – it is particularly interesting how little was done concerning the theory of games by practitioners in other fields. Any understanding of why applied game theory is most common in economics surely must be able to explain the absolute dominance of economists in pure game theory. Second, here is as good a place as any to give some dap to my home department, Northwestern MEDS. Some incredible amount of the important work in this game theory golden age was done there. To name only a few, the department at one time was home to Milgrom, Myerson, Stokey, Satterthwaite, Holmstrom, Baron, Ledyard, Kalai, Kamien, Sonnenschein, John Roberts, almost all of whom were young researchers at their productive peaks – ex-post, that must have been the best theoretical economics department of all time, right?

Final article from Theory and Society (No IDEAS link found). Tips of the hat to Richard Swedberg for posting full, final articles on his website, and to the post at orgtheory where I first came across this article.

“Mathematical Models in the Social Sciences,” K. Arrow (1951)

I have given Paul Samuelson the title of “greatest economist ever” many times on this site. If he is number one, though, Ken Arrow is surely second. And this essay, an early Cowles discussion paper, is an absolute must-read.

Right on the first page is an absolute destruction of every ridiculous statement you’ve ever heard about mathematical economics. Quoting the physicist Gibbs: “Mathematics is a language.” On whether quantitative methods are appropriate for studying human action: “Doubtless many branches of mathematics – especially those most familiar to the average individual, such as algebra and the calculus – are quantitative in nature. But the whole field of symbolic logic is purely qualitative. We can frame such questions as the following: Does the occurrence of one event imply the occurrence of another? Is it impossible that two events should both occur?” This is spot on. What is most surprising to me, wearing my theorist hat, is how little twentieth century mathematics occurs in economics vis-a-vis the pure sciences, not how much. The most prominent mathematics in economics are the theories of probability, various forms of mathematical logic, and existence theorems on wholly abstract spaces, meaning spaces that don’t have any obvious correspondence with the physical world. These techniques tell us little about numbers, but rather help us answer questions like “How does X relate to Y?” and “Is Z a logical possibility?” and “For some perhaps unknown sets of beliefs, how serious a problem can Q cause?” All of these statements look to me to be exactly equivalent to the types of a priori logical reasoning which appear everywhere in 18th and 19th century “nonmathematical” social science.

There is a common objection to mathematical theorizing, that mathematics is limited in nature compared to the directed intuition which a good social scientist can verbalize. This is particularly true compared to the pure sciences. We have very little intuition about atoms, but great intuition about the social world we inhabit. Arrow argues, however, that making valid logical implication is a difficult task indeed, particularly if we’re using any deductive reasoning beyond the simplest tools in Aristotle. Writing our verbal thoughts as mathematics allows the use of more complicated deductive tools. And the same is true of induction: mathematical model building allows for the use of (what was then very modern) statistical tools to identify relationships. Naive regression identifies correlations, but is rarely able to discuss any more complex relationship between data.

A final note: if you’re interested in history of thought, there are some interesting discussions of decision theory pre-Savage and game theory pre-Nash and pre-Harsanyi in Arrow’s article. A number of interpretations are given that seem somewhat strange given our current understanding, such as interpreting mixed strategies as “bluffing,” or writing down positive-sum n-person cooperative games as zero-sum n+1 player games where a “fictitious player” eats the negative outcome. Less strange, but still by no means mainstream, is Wald’s interpretation of statistical inference as a zero-sum game against nature, where the statistician with a known loss function chooses a decision function (perhaps mixed) and nature simultaneously chooses a realization in order to maximize the expected loss. There is an interesting discussion of something that looks an awful lot like evolutionary game theory, proposed by Nicholas Rachevsky in 1947; I hadn’t known these non-equilibrium linear ODE games existed that far before Maynard Smith. Arrow, and no doubt his contemporaries, also appear to have been quite optimistic about the possibility of a dynamic game theory that incorporated learning about opponent’s play axiomatically, but I would say that, in 2012, we have no such theory and for a variety of reasons, a suitable one may not be possible. Finally, Arrow notes an interesting discussion between Koopmans and his contemporaries about methodological individualism; Arrow endorses the idea that, would we have the data, society’s aggregate outcomes are necessarily determined wholly by the actions of individuals. There is no “societal organism”. Many economists, no doubt, agree completely with that statement, though there are broad groups in the social sciences who both think that the phrase “would we have the data” is a more serious concern that economists generally consider it, and that conceive of non-human social actors. It’s worthwhile to at least know these arguments are out there.

http://128.36.236.35/P/cp/p00a/p0048.pdf (Final version provided thanks to the Cowles Commission’s lovely open access policy)

“Note on the Theory of the Economy of Research,” C. S. Peirce (1879)

Though this site is devoted generally to new research, the essay discussed in this post, I trust, will be new enough to the vast majority of readers. Charles Sanders Peirce is a titan of analytic philosophy, and there is certainly a case to be made that he is the greatest American philosopher of all time. He also has had a fairly well-known indirect influence on economics: Peirce was in some ways rediscovered by the great mathematician Alfred Tarski, who then taught Kenneth Arrow, and in doing so may have introduced Peirce’s relational algebra to the field of economics. (You may be thinking, relational algebra, what is that? But you certainly know what it is: take a set, apply a perhaps partial, often binary ordering with certain properties, then prove results. This surely describes every modern introduction to the theory of preferences, does it not?) But Peirce also has an essay more directly on economics that is fascinating to see in retrospect. This Peirce essay is reprinted in Phil Mirowski’s book “Science Bought and Sold” along with notes on the essay by James Wible which I shall also draw from.

Two final things. First, I note, if only to myself, the following quote from Peirce to be used in a future research paper of my own: “Economical science is particularly profitable to science; and that of all the branches of economy, the economy of research is the most profitable.” Second, check out where this essay was published: the annual report of the U.S. government Coast Survey of 1879! No wonder it has been overlooked. If you know anything of the biography of Peirce, though, there is not much surprising in this odd location. Peirce was supposedly such a nut that, despite obvious brilliance, he was repeatedly blackballed from academic appointments by future colleagues around the country!

Wible claims, and I also know of no earlier such work, that this Peirce essay is the earliest mathematical work on the theory of invention. And given the intellectual history, this seems almost certain to be so. The essay was written right at the cusp of the marginal revolution and mathematical political economy, Peirce is known to have been familiar with the few scraps of earlier mathematical economics like Cournot’s famous 1838 essay, and Peirce is the father of a philosophical school for which selecting the best line of research to examine in order to learn inductively was a pressing concern. If you’ve ever read economics articles from the middle of the 19th century, this one will shock you: in style, I think it is essentially publishable today. It looks like 21st century economics. There are marginal tradeoffs. There is social science done by mathematical manipulation of heavily abstracted concepts. There is even a Marshallian diagram! It’s phenomenal. Since this looks like modern economics, let’s discuss it like modern economics; what does Peirce’s theory say?

As he introduced it, “I considered this problem. Somebody furnished a fund to be expended upon research without restrictions. What sort of researches should it be expended upon?” Essentially, there are some scientific problems which we understand only vaguely; you may think of this purely qualitatively, or as meaning something is measured to within some confidence interval. There are diminishing returns to science, so that while decreasing error can be done at linear cost, the utility gained from such reduction is concave (the inverse is quadratic in Peirce’s formulation). There is a total fixed research budget. What should be worked on first? Note that this paper was first written in 1876: there is no stochastic learning or any such thing, as the mathematics to discuss bandits and related objects was not yet developed. Learning is purely deterministic here.

Solving that constrained maximization problem gives the now-familiar, but then-nonexistent, result that we should compare ratios of MU/MC across different projects. Peirce called this ratio of marginal utility to marginal cost the “economic urgency” of a given line of research. He notes that, given that functional form assumptions, new research fields where we know very little are particularly worthwhile investments: the gains from increasing our knowledge are exponential in ignorance, whereas the cost is linear. As an example, an early chemist with simple vials is able to provide results with more social utility than a thousand chemists working in Peirce’s day with all sorts of modern equipment. Peirce also derives a result concerning sampling which is a bit opaque for modern readers given that it is couched in terms of “accidental probable error” rather than confidence intervals; nonetheless, it is very Wald-esque in that it explicitly argues that optimal sample size in experiments depends crucially on the budget, the costs of sampling and the utility of learning inferences from that sampling. Such considerations are absolutely ignored in a lot of research design even today!

http://books.google.com/books?id=ux79s_IhpFYC (Both Peirce’s original essay and Wible’s commentary appear in “Science Bought and Sold,” edited by Mirowski and Sent. The Google Books Preview is generous enough here for you to read the entirety of both essays; I do not see any other ungated copies of either online.)

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