Category Archives: History of Economic Thought

The Economics of John Nash

I’m in the midst of a four week string of conferences and travel, and terribly backed up with posts on some great new papers, but I can’t let the tragic passing today of John Nash go by without comment. When nonacademics ask what I do, I often say that I work in a branch of applied math called game theory; if you say you are economist, the man on the street expects you to know when unemployment will go down, or which stocks they should buy, or whether monetary expansion will lead to inflation, questions which the applied theorist has little ability to answer in a satisfying way. But then, if you mention game theory, without question the most common way your interlocutor knows the field is via Russell Crowe’s John Nash character in A Beautiful Mind, so surely, and rightfully, no game theorist has greater popular name recognition.

Now Nash’s contributions to economics are very small, though enormously influential. He was a pure mathematician who took only one course in economics in his studies; more on this fortuitous course shortly. The contributions are simple to state: Nash founded the theory of non-cooperative games, and he instigated an important, though ultimately unsuccessful, literature on bargaining. Nash essentially only has two short papers on each topic, each of which is easy to follow for a modern reader, so I will generally discuss some background on the work rather than the well-known results directly.

First, non-cooperative games. Robert Leonard has a very interesting intellectual history of the early days of game theory, the formal study of strategic interaction, which begins well before Nash. Many like to cite von Neumann’s “Zur Theorie der Gesellschaftsspiele” (“A Theory of Parlor Games”), from whence we have the minimax theorem, but Emile Borel in the early 1920’s, and Ernst Zermelo with his eponymous theorem a decade earlier, surely form relevant prehistory as well. These earlier attempts, including von Neumann’s book with Morganstern, did not allow general investigation of what we now call noncooperative games, or strategic situations where players do not attempt to collude. The most famous situation of this type is the Prisoner’s Dilemma, a simple example, yet a shocking one: competing agents, be they individuals, firms or countries, may (in a sense) rationally find themselves taking actions which both parties think is worse than some alternative. Given the U.S. government interest in how a joint nuclear world with the Soviets would play out, analyzing situations of that type was not simply a “Gesellschaftsspiele” in the late 1940s; Nash himself was funded by the Atomic Energy Commission, and RAND, site of a huge amount of important early game theory research, was linked to the military.

Nash’s insight was, in retrospect, very simple. Consider a soccer penalty kick, where the only options are to kick left and right for the shooter, and to simultaneously dive left or right for the goalie. Now at first glance, it seems like there can be no equilibrium: if the shooter will kick left, then the goalie will jump to that side, in which case the shooter would prefer to shoot right, in which case the goalie would prefer to switch as well, and so on. In real life, then, what do we expect to happen? Well, surely we expect that the shooter will sometimes shoot left and sometimes right, and likewise the goalie will mix which way she dives. That is, instead of two strategies for each player, we have a continuum of mixed strategies, where a mixed strategy is simply a probability distribution over the strategies “Left, Right”. This idea of mixed strategies “convexifies” the strategy space so that we can use fixed point strategies to guarantee that an equilibrium exists in every finite-strategy noncooperative game under expected utility (Kakutani’s Fixed Point in the initial one-page paper in PNAS which Nash wrote his very first year of graduate school, and Brouwer’s Fixed Point in the Annals of Math article which more rigorously lays out Nash’s noncooperative theory). Because of Nash, we are able to analyze essentially whatever nonstrategic situation we want under what seems to be a reasonable solution concept (I optimize given my beliefs about what others will do, and my beliefs are in the end correct). More importantly, the fixed point theorems Nash used to generate his equilibria are now so broadly applied that no respectable economist should now get a PhD without understanding how they work.

(A quick aside: it is quite interesting to me that game theory, as opposed to Walrasian/Marshallian economics, does not derive from physics or other natural sciences, but rather from a program at the intersection of formal logic and mathematics, primarily in Germany, primarily in the early 20th century. I still have a mind to write a proper paper on this intellectual history at some point, but there is a very real sense in which economics post-Samuelson, von Neumann and Nash forms a rather continuous methodology with earlier social science in the sense of qualitative deduction, whereas it is our sister social sciences which, for a variety of reasons, go on writing papers without the powerful tools of modern logic and the mathematics which followed Hilbert. When Nash makes claims about the existence of equilibria due to Brouwer, the mathematics is merely the structure holding up and extending ideas concerning the interaction of agents in noncooperative systems that would have been totally familiar to earlier generations of economists who simply didn’t have access to tools like the fixed point theorems, in the same way that Samuelson and Houthakker’s ideas on utility are no great break from earlier work aside from their explicit incorporation of deduction on the basis of relational logic, a tool unknown to economists in the 19th century. That is, I claim the mathematization of economics in the mid 20th century represents no major methodological break, nor an attempt to ape the natural sciences. Back to Nash’s work in the direct sense.)

Nash only applies his theory to one game: a simplified version of poker due to his advisor called Kuhn Poker. It turned out that the noncooperative solution was not immediately applicable, at least to the types of applied situations where it is now commonplace, without a handful of modifications. In my read of the intellectual history, noncooperative games was a bit of a failure outside the realm of pure math in its first 25 years because we still needed Harsanyi’s purification theorem and Bayesian equilibria to understand what exactly was going on with mixed strategies, Reinhard Selten’s idea of subgame perfection to reasonably analyze games with multiple stages, and the idea of mechanism design of Gibbard, Vickers, Myerson, Maskin, and Satterthwaite (among others) to make it possible to discuss how institutions affect outcomes which are determined in equilibrium. It is not simply economists that Nash influenced; among many others, his work directly leads to the evolutionary games of Maynard Smith and Price in biology and linguistics, the upper and lower values of his 1953 results have been used to prove other mathematical results and to discuss what is meant as truth in philosophy, and Nash is widespread in the analysis of voting behavior in political science and international relations.

The bargaining solution is a trickier legacy. Recall Nash’s sole economics course, which he took as an undergraduate. In that course, he wrote a term paper, eventually to appear in Econometrica, where he attempted to axiomatize what will happen when two parties bargain over some outcome. The idea is simple. Whatever the bargaining outcome is, we want it to satisfy a handful of reasonable assumptions. First, since ordinal utility is invariant to affine transformations of a utility function, the bargaining outcome should not be affected by these types of transformations: only ordinal preferences should matter. Second, the outcome should be Pareto optimal: the players would have to mighty spiteful to throw away part of the pie rather than give it to at least one of them. Third, given their utility functions players should be treated symmetrically. Fourth (and a bit controversially, as we will see), Nash insisted on Independence of Irrelevant Alternatives, meaning that if f(T) is the set of “fair bargains” when T is the set of all potential bargains, then if the potential set of bargains is smaller yet still contains f(T), say S strictly contained by T where f(T) is in S, then f(T) must remain the barganing outcome. It turns out that under these assumptions, there is a unique outcome which maximizes (u(x)-u(d))*(v(x)-v(d)), where u and v are each player’s utility functions, x is the vector of payoffs under the eventual bargain, and d the “status-quo” payoff if no bargain is made. This is natural in many cases. For instance, if two identical agents are splitting a dollar, then 50-50 is the only Nash outcome. Uniqueness is not at all obvious: recall the Edgeworth box and you will see that individual rationality and Pareto optimality alone leave many potential equilibria. Nash’s result is elegant and surprising, and it is no surprise that Nash’s grad school recommendation letter famously was only one sentence long: “This man is a genius.”

One problem with Nash bargaining, however. Nash was famously bipolar in real life, but there is an analogous bipolar feel to the idea of Nash equilibrium and the idea of Nash bargaining: where exactly are threats in Nash’s bargain theory? That is, Nash bargaining as an idea completely follows from the cooperative theory of von Neumann and Morganstern. Consider two identical agents splitting a dollar once more. Imagine that one of the agents already has 30 cents, so that only 70 of the cents are actually in the middle of the table. The Nash solution is that the person who starts with the thirty cents eventually winds up with 65 cents, and the other person with 35. But play this out in your head.

Player 1: “I, already having the 30 cents, should get half of what remains. It is only fair, and if you don’t give me 65 I will walk away from this table and we will each get nothing more.”

Player 2: “What does that have to do with it? The fair outcome is 50 cents each, which leaves you with more than your originally thirty, so you can take your threat and go jump off a bridge.”

That is, 50/50 might be a reasonable solution here, right? This might make even more sense if we take a more concrete example: bargaining over wages. Imagine the prevailing wage for CEOs in your industry is $250,000. Two identical CEOs will generate $500,000 in value for the firm if hired. CEO Candidate One has no other job offer. CEO Candidate Two has an offer from a job with similar prestige and benefits, paying $175,000. Surely we can’t believe that the second CEO will wind up with higher pay, right? It is a completely noncredible threat to take the $175,000 offer, hence it shouldn’t affect the bargaining outcome. A pet peeve of mine is that many applied economists are still using Nash bargaining – often in the context of the labor market! – despite this well-known problem.

Nash was quite aware of this, as can be seen by his 1953 Econometrica, where he attempts to give a noncooperative bargaining game that reaches the earlier axiomatic outcome. Indeed, this paper inspired an enormous research agenda called the Nash Program devoted to finding noncooperative games which generate well-known or reasonable-sounding cooperative solution outcomes. In some sense, the idea of “implementation” in mechanism design, where we investigate whether there exists a game which can generate socially or coalitionally preferred outcomes noncooperatively, can be thought of as a successful modern branch of the Nash program. Nash’s ’53 noncooperative game simply involves adding a bit of noise into the set of possible outcomes. Consider splitting a dollar again. Let a third party tell each player to name how many cents they want. If the joint requests are feasible, then the dollar is split (with any remainder thrown away), else each player gets nothing. Clearly every split of the dollar on the Pareto frontier is a Nash equilibrium, as is each player requesting the full dollar and getting nothing. However, if there is a tiny bit of noise about whether there is exactly one dollar, or .99 cents, or 1.01 cents, etc., when deciding whether to ask for more money, I will have to weigh the higher payoff if the joint demand is feasible against the payoff zero if my increased demand makes the split impossible and hence neither of us earn anything. In a rough sense, Nash shows that as the distribution of noise becomes degenerate around the true bargaining frontier, players will demand exactly their Nash bargaining outcome. Of course it is interesting that there exists some bargaining game that generates the Nash solution, and the idea that we should study noncooperative games which implement cooperate solution concepts is without a doubt seminal, but this particular game seems very strange to me, as I don’t understand what the source of the noise is, why it becomes degenerate, etc.

On the shoulders of Nash, however, bargaining progressed a huge amount. Three papers in particular are worth your time, although hopefully you have seen these before: Kalai and Smorodinsky 1975 who retaining the axiomatic approach but drop IIA, Rubinstein’s famous 1982 Econometrica on noncooperative bargaining with alternative offers, and Binmore, Rubinstein and Wolinsky on implementation of bargaining solutions which deals with the idea of threats as I did above.

You can read all four Nash papers in the original literally during your lunch hour; this seems to me a worthy way to tip your cap toward a man who literally helped make modern economics possible.

“The Contributions of the Economics of Information to Twentieth Century Economics,” J. Stiglitz (2000)

There have been three major methodological developments in economics since 1970. First, following the Lucas Critique we are reluctant to accept policy advice which is not the result of directed behavior on the part of individuals and firms. Second, developments in game theory have made it possible to reformulate questions like “why do firms exist?”, “what will result from regulating a particular industry in a particular way?”, “what can I infer about the state of the world from an offer to trade?”, among many others. Third, imperfect and asymmetric information was shown to be of first-order importance for analyzing economic problems.

Why is information so important? Prices, Hayek taught us, solve the problem of asymmetric information about scarcity. Knowing the price vector is a sufficient statistic for knowing everything about production processes in every firm, as far as generating efficient behavior is concerned. The simple existence of asymmetric information, then, is not obviously a problem for economic efficiency. And if asymmetric information about big things like scarcity across society does not obviously matter, then how could imperfect information about minor things matter? A shopper, for instance, may not know exactly the price of every car at every dealership. But “Natura non facit saltum”, Marshall once claimed: nature does not make leaps. Tiny deviations from the assumptions of general equilibrium do not have large consequences.

But Marshall was wrong: nature does make leaps when it comes to information. The search model of Peter Diamond, most famously, showed that arbitrarily small search costs lead to firms charging the monopoly price in equilibrium, hence a welfare loss completely out of proportion to the search costs. That is, information costs and asymmetries, even very small ones, can theoretically be very problematic for the Arrow-Debreu welfare properties.

Even more interesting, we learned that prices are more powerful than we’d believed. They convey information about scarcity, yes, but also information about other people’s own information or effort. Consider, for instance, efficiency wages. A high wage is not merely a signal of scarcity for a particular type of labor, but is simultaneously an effort inducement mechanism. Given this dual role, it is perhaps not surprising that general equilibrium is no longer Pareto optimal, even if the planner is as constrained informationally as each agent.

How is this? Decentralized economies may, given information cost constraints, exert too much effort searching, or generate inefficient separating equilibrium that unravel trades. The beautiful equity/efficiency separation of the Second Welfare Theorem does not hold in a world of imperfect information. A simple example on this point is that it is often useful to allow some agents suffering moral hazard worries to “buy the firm”, mitigating the incentive problem, but limited liability means this may not happen unless those particular agents begin with a large endowment. That is, a different endowment, where the agents suffering extreme moral hazard problems begin with more money and are able to “buy the firm”, leads to more efficient production (potentially in a Pareto sense) than an endowment where those workers must be provided with information rents in an economy-distorting manner.

It is a strange fact that many social scientists feel economics to some extent stopped progressing by the 1970s. All the important basic results were, in some sense, known. How untrue this is! Imagine labor without search models, trade without monopolistic competitive equilibria, IO or monetary policy without mechanism design, finance without formal models of price discovery and equilibrium noise trading: all would be impossible given the tools we had in 1970. The explanations that preceded modern game theoretic and information-laden explanations are quite extraordinary: Marshall observed that managers have interests different from owners, yet nonetheless are “well-behaved” in running firms in a way acceptable to the owner. His explanation was to credit British upbringing and morals! As Stiglitz notes, this is not an explanation we would accept today. Rather, firms have used a number of intriguing mechanisms to structure incentives in a way that limits agency problems, and we now possess the tools to analyze these mechanisms rigorously.

Final 2000 QJE (RePEc IDEAS)

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.

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