Category Archives: History of Economic Thought

Nobel Prize 2016 Part II: Oliver Hart

The Nobel Prize in Economics was given yesterday to two wonderful theorists, Bengt Holmstrom and Oliver Hart. I wrote a day ago about Holmstrom’s contributions, many of which are simply foundational to modern mechanism design and its applications. Oliver Hart’s contribution is more subtle and hence more of a challenge to describe to a nonspecialist; I am sure of this because no concept gives my undergraduate students more headaches than Hart’s “residual control right” theory of the firm. Even stranger, much of Hart’s recent work repudiates the importance of his most famous articles, a point that appears to have been entirely lost on every newspaper discussion of Hart that I’ve seen (including otherwise very nice discussions like Applebaum’s in the New York Times). A major reason he has changed his beliefs, and his research agenda, so radically is not simply the whims of age or the pressures of politics, but rather the impact of a devastatingly clever, and devastatingly esoteric, argument made by the Nobel winners Eric Maskin and Jean Tirole. To see exactly what’s going on in Hart’s work, and why there remains many very important unsolved questions in this area, let’s quickly survey what economists mean by “theory of the firm”.

The fundamental strangeness of firms goes back to Coase. Markets are amazing. We have wonderful theorems going back to Hurwicz about how competitive market prices coordinate activity efficiently even when individuals only have very limited information about how various things can be produced by an economy. A pencil somehow involves graphite being mined, forests being explored and exploited, rubber being harvested and produced, the raw materials brought to a factory where a machine puts the pencil together, ships and trains bringing the pencil to retail stores, and yet this decentralized activity produces a pencil costing ten cents. This is the case even though not a single individual anywhere in the world knows how all of those processes up the supply chain operate! Yet, as Coase pointed out, a huge amount of economic activity (including the majority of international trade) is not coordinated via the market, but rather through top-down Communist-style bureaucracies called firms. Why on Earth do these persistent organizations exist at all? When should firms merge and when should they divest themselves of their parts? These questions make up the theory of the firm.

Coase’s early answer is that something called transaction costs exist, and that they are particularly high outside the firm. That is, market transactions are not free. Firm size is determined at the point where the problems of bureaucracy within the firm overwhelm the benefits of reducing transaction costs from regular transactions. There are two major problems here. First, who knows what a “transaction cost” or a “bureaucratic cost” is, and why they differ across organizational forms: the explanation borders on tautology. Second, as the wonderful paper by Alchian and Demsetz in 1972 points out, there is no reason we should assume firms have some special ability to direct or punish their workers. If your supplier does something you don’t like, you can keep them on, or fire them, or renegotiate. If your in-house department does something you don’t like, you can keep them on, or fire them, or renegotiate. The problem of providing suitable incentives – the contracting problem – does not simply disappear because some activity is brought within the boundary of the firm.

Oliver Williamson, a recent Nobel winner joint with Elinor Ostrom, has a more formal transaction cost theory: some relationships generate joint rents higher than could be generated if we split ways, unforeseen things occur that make us want to renegotiate our contract, and the cost of that renegotiation may be lower if workers or suppliers are internal to a firm. “Unforeseen things” may include anything which cannot be measured ex-post by a court or other mediator, since that is ultimately who would enforce any contract. It is not that everyday activities have different transaction costs, but that the negotiations which produce contracts themselves are easier to handle in a more persistent relationship. As in Coase, the question of why firms do not simply grow to an enormous size is largely dealt with by off-hand references to “bureaucratic costs” whose nature was largely informal. Though informal, the idea that something like transaction costs might matter seemed intuitive and had some empirical support – firms are larger in the developing world because weaker legal systems means more “unforeseen things” will occur outside the scope of a contract, hence the differential costs of holdup or renegotiation inside and outside the firm are first order when deciding on firm size. That said, the Alchian-Demsetz critique, and the question of what a “bureaucratic cost” is, are worrying. And as Eric van den Steen points out in a 2010 AER, can anyone who has tried to order paper through their procurement office versus just popping in to Staples really believe that the reason firms exist is to lessen the cost of intrafirm activities?

Grossman and Hart (1986) argue that the distinction that really makes a firm a firm is that it owns assets. They retain the idea that contracts may be incomplete – at some point, I will disagree with my suppliers, or my workers, or my branch manager, about what should be done, either because a state of the world has arrived not covered by our contract, or because it is in our first-best mutual interest to renegotiate that contract. They retain the idea that there are relationship-specific rents, so I care about maintaining this particular relationship. But rather than rely on transaction costs, they simply point out that the owner of the asset is in a much better bargaining position when this disagreement occurs. Therefore, the owner of the asset will get a bigger percentage of rents after renegotiation. Hence the person who owns an asset should be the one whose incentive to improve the value of the asset is most sensitive to that future split of rents.

Baker and Hubbard (2004) provide a nice empirical example: when on-board computers to monitor how long-haul trucks were driven began to diffuse, ownership of those trucks shifted from owner-operators to trucking firms. Before the computer, if the trucking firm owns the truck, it is hard to contract on how hard the truck will be driven or how poorly it will be treated by the driver. If the driver owns the truck, it is hard to contract on how much effort the trucking firm dispatcher will exert ensuring the truck isn’t sitting empty for days, or following a particularly efficient route. The computer solves the first problem, meaning that only the trucking firm is taking actions relevant to the joint relationship which are highly likely to be affected by whether they own the truck or not. In Grossman and Hart’s “residual control rights” theory, then, the introduction of the computer should mean the truck ought, post-computer, be owned by the trucking firm. If these residual control rights are unimportant – there is no relationship-specific rent and no incompleteness in contracting – then the ability to shop around for the best relationship is more valuable than the control rights asset ownership provides. Hart and Moore (1990) extends this basic model to the case where there are many assets and many firms, suggesting critically that sole ownership of assets which are highly complementary in production is optimal. Asset ownership affects outside options when the contract is incomplete by changing bargaining power, and splitting ownership of complementary assets gives multiple agents weak bargaining power and hence little incentive to invest in maintaining the quality of, or improving, the assets. Hart, Schleifer and Vishny (1997) provide a great example of residual control rights applied to the question of why governments should run prisons but not garbage collection. (A brief aside: note the role that bargaining power plays in all of Hart’s theories. We do not have a “perfect” – in a sense that can be made formal – model of bargaining, and Hart tends to use bargaining solutions from cooperative game theory like the Shapley value. After Shapley’s prize alongside Roth a few years ago, this makes multiple prizes heavily influenced by cooperative games applied to unexpected problems. Perhaps the theory of cooperative games ought still be taught with vigor in PhD programs!)

There are, of course, many other theories of the firm. The idea that firms in some industries are big because there are large fixed costs to enter at the minimum efficient scale goes back to Marshall. The agency theory of the firm going back at least to Jensen and Meckling focuses on the problem of providing incentives for workers within a firm to actually profit maximize; as I noted yesterday, Holmstrom and Milgrom’s multitasking is a great example of this, with tasks being split across firms so as to allow some types of workers to be given high powered incentives and others flat salaries. More recent work by Bob Gibbons, Rebecca Henderson, Jon Levin and others on relational contracting discusses how the nexus of self-enforcing beliefs about how hard work today translates into rewards tomorrow can substitute for formal contracts, and how the credibility of these “relational contracts” can vary across firms and depend on their history.

Here’s the kicker, though. A striking blow was dealt to all theories which rely on the incompleteness or nonverifiability of contracts by a brilliant paper of Maskin and Tirole (1999) in the Review of Economic Studies. Theories relying on incomplete contracts generally just hand-waved that there are always events which are unforeseeable ex-ante or impossible to verify in court ex-post, and hence there will always scope for disagreement about what to do when those events occur. But, as Maskin and Tirole correctly point out, agent don’t care about anything in these unforeseeable/unverifiable states except for what the states imply about our mutual valuations from carrying on with a relationship. Therefore, every “incomplete contract” should just involve the parties deciding in advance that if a state of the world arrives where you value keeping our relationship in that state at 12 and I value it at 10, then we should split that joint value of 22 at whatever level induces optimal actions today. Do this same ex-ante contracting for all future profit levels, and we are done. Of course, there is still the problem of ensuring incentive compatibility – why would the agents tell the truth about their valuations when that unforeseen event occurs? I will omit the details here, but you should read the original paper where Maskin and Tirole show a (somewhat convoluted but still working) mechanism that induces truthful revelation of private value by each agent. Taking the model’s insight seriously but the exact mechanism less seriously, the paper basically suggests that incomplete contracts don’t matter if we can truthfully figure out ex-post who values our relationship at what amount, and there are many real-world institutions like mediators who do precisely that. If, as Maskin and Tirole prove (and Maskin described more simply in a short note), incomplete contracts aren’t a real problem, we are back to square one – why have persistent organizations called firms?

What should we do? Some theorists have tried to fight off Maskin and Tirole by suggesting that their precise mechanism is not terribly robust to, for instance, assumptions about higher-order beliefs (e.g., Aghion et al (2012) in the QJE). But these quibbles do not contradict the far more basic insight of Maskin and Tirole, that situations we think of empirically as “hard to describe” or “unlikely to occur or be foreseen”, are not sufficient to justify the relevance of incomplete contracts unless we also have some reason to think that all mechanisms which split rent on the basis of future profit, like a mediator, are unavailable. Note that real world contracts regularly include provisions that ex-ante describe how contractual disagreement ex-post should be handled.

Hart’s response, and this is both clear from his CV and from his recent papers and presentations, is to ditch incompleteness as the fundamental reason firms exist. Hart and Moore’s 2007 AER P&P and 2006 QJE are very clear:

Although the incomplete contracts literature has generated some useful insights about firm boundaries, it has some shortcomings. Three that seem particularly important to us are the following. First, the emphasis on noncontractible ex ante investments seems overplayed: although such investments are surely important, it is hard to believe that they are the sole drivers of organizational form. Second, and related, the approach is ill suited to studying the internal organization of firms, a topic of great interest and importance. The reason is that the Coasian renegotiation perspective suggests that the relevant parties will sit down together ex post and bargain to an efficient outcome using side payments: given this, it is hard to see why authority, hierarchy, delegation, or indeed anything apart from asset ownership matters. Finally, the approach has some foundational weaknesses [pointed out by Maskin and Tirole (1999)].

To my knowledge, Oliver Hart has written zero papers since Maskin-Tirole was published which attempt to explain any policy or empirical fact on the basis of residual control rights and their necessary incomplete contracts. Instead, he has been primarily working on theories which depend on reference points, a behavioral idea that when disagreements occur between parties, the ex-ante contracts are useful because they suggest “fair” divisions of rent, and induce shading and other destructive actions when those divisions are not given. These behavioral agents may very well disagree about what the ex-ante contract means for “fairness” ex-post. The primary result is that flexible contracts (e.g., contracts which deliberately leave lots of incompleteness) can adjust easily to changes in the world but will induce spiteful shading by at least one agent, while rigid contracts do not permit this shading but do cause parties to pursue suboptimal actions in some states of the world. This perspective has been applied by Hart to many questions over the past decade, such as why it can be credible to delegate decision making authority to agents; if you try to seize it back, the agent will feel aggrieved and will shade effort. These responses are hard, or perhaps impossible, to justify when agents are perfectly rational, and of course the Maskin-Tirole critique would apply if agents were purely rational.

So where does all this leave us concerning the initial problem of why firms exist in a sea of decentralized markets? In my view, we have many clever ideas, but still do not have the perfect theory. A perfect theory of the firm would need to be able to explain why firms are the size they are, why they own what they do, why they are organized as they are, why they persist over time, and why interfirm incentives look the way they do. It almost certainly would need its mechanisms to work if we assumed all agents were highly, or perfectly, rational. Since patterns of asset ownership are fundamental, it needs to go well beyond the type of hand-waving that makes up many “resource” type theories. (Firms exist because they create a corporate culture! Firms exist because some firms just are better at doing X and can’t be replicated! These are outcomes, not explanations.) I believe that there are reasons why the costs of maintaining relationships – transaction costs – endogenously differ within and outside firms, and that Hart is correct is focusing our attention on how asset ownership and decision making authority affects incentives to invest, but these theories even in their most endogenous form cannot do everything we wanted a theory of the firm to accomplish. I think that somehow reputation – and hence relational contracts – must play a fundamental role, and that the nexus of conflicting incentives among agents within an organization, as described by Holmstrom, must as well. But we still lack the precise insight to clear up this muddle, and give us a straightforward explanation for why we seem to need “little Communist bureaucracies” to assist our otherwise decentralized and almost magical market system.

Nobel Prize 2016 Part I: Bengt Holmstrom

The Nobel Prize in Economics has been announced, and what a deserving prize it is: Bengt Holmstrom and Oliver Hart have won for the theory of contracts. The name of this research weblog is “A Fine Theorem”, and it would be hard to find two economists whose work is more likely to elicit such a description! Both are incredibly deserving; more than five years ago on this site, I discussed how crazy it was that Holmstrom had yet to win!. The only shock is the combination: a more natural prize would have been Holmstrom with Paul Milgrom and Robert Wilson for modern applied mechanism design, and Oliver Hart with John Moore and Sandy Grossman for the theory of the firm. The contributions of Holmstrom and Hart are so vast that I’m splitting this post into two, so as to properly cover the incredible intellectual accomplishments of these two economists.

The Finnish economist Bengt Holmstrom did his PhD in operations research at Stanford, advised by Robert Wilson, and began his career at my alma mater, the tiny department of Managerial Economics and Decision Sciences at Northwestern’s Kellogg School. To say MEDS struck gold with their hires in this era is an extreme understatement: in 1978 and 1979 alone, they hired Holmstrom and his classmate Paul Milgrom (another Wilson student from Stanford), hired Nancy Stokey promoted Nobel laureate Roger Myerson to Associate Professor, and tenured an adviser of mine, Mark Satterthwaite. And this list doesn’t even include other faculty in the late 1970s and early 1980s like eminent contract theorist John Roberts, behavioralist Colin Camerer, mechanism designer John Ledyard or game theorist Ehud Kalai. This group was essentially put together by two senior economists at Kellogg, Nancy Schwartz and Stanley Reiter, who had the incredible foresight to realize both that applied game theory was finally showing promise of tackling first-order economic questions in a rigorous way, and that the folks with the proper mathematical background to tackle these questions were largely going unhired since they often did their graduate work in operations or mathematics departments rather than traditional economics departments. This market inefficiency, as it were, allowed Nancy and Stan to hire essentially every young scholar in what would become the field of mechanism design, and to develop a graduate program which combined operations, economics, and mathematics in a manner unlike any other place in the world.

From that fantastic group, Holmstrom’s contribution lies most centrally in the area of formal contract design. Imagine that you want someone – an employee, a child, a subordinate division, an aid contractor, or more generally an agent – to perform a task. How should you induce them to do this? If the task is “simple”, meaning the agent’s effort and knowledge about how to perform the task most efficiently is known and observable, you can simply pay a wage, cutting off payment if effort is not being exerted. When only the outcome of work can be observed, if there is no uncertainty in how effort is transformed into outcomes, knowing the outcome is equivalent to knowing effort, and hence optimal effort can be achieved via a bonus payment made on the basis of outcomes. All straightforward so far. The trickier situations, which Holmstrom and his coauthors analyzed at great length, are when neither effort nor outcomes are directly observable.

Consider paying a surgeon. You want to reward the doctor for competent, safe work. However, it is very difficult to observe perfectly what the surgeon is doing at all times, and basing pay on outcomes has a number of problems. First, the patient outcome depends on the effort of not just one surgeon, but on others in the operating room and prep table: team incentives must be provided. Second, the doctor has many ways to shift the balance of effort between reducing costs to the hospital, increasing patient comfort, increasing the quality of the medical outcome, and mentoring young assistant surgeons, so paying on the basis of one or two tasks may distort effort away from other harder-to-measure tasks: there is a multitasking problem. Third, the number of medical mistakes, or the cost of surgery, that a hospital ought expect from a competent surgeon depends on changes in training and technology that are hard to know, and hence a contract may want to adjust payments for its surgeons on the performance of surgeons elsewhere: contracts ought take advantage of relevant information when it is informative about the task being incentivized. Fourth, since surgeons will dislike risk in their salary, the fact that some negative patient outcomes are just bad luck means that you will need to pay the surgeon very high bonuses to overcome their risk aversion: when outcome measures involve uncertainty, optimal contracts will weigh “high-powered” bonuses against “low-powered” insurance against risk. Fifth, the surgeon can be incentivized either by payments today or by keeping their job tomorrow, and worse, these career concerns may cause the surgeon to waste the hospital’s money on tasks which matter to the surgeon’s career beyond the hospital.

Holmstrom wrote the canonical paper on each of these topics. His 1979 paper in the Bell Journal of Economics shows that any information which reduces the uncertainty about what an agent actually did should feature in a contract, since by reducing uncertainty, you reduce the risk premium needed to incentivize the agent to accept the contract. It might seem strange that contracts in many cases do not satisfy this “informativeness principle”. For instance, CEO bonuses are often not indexed to the performance of firms in the same industry. If oil prices rise, essentially all oil firms will be very profitable, and this is true whether or not a particular CEO is a good one. Bertrand and Mullainathan argue that this is because many firms with diverse shareholders are poorly governed!

The simplicity of contracts in the real world may have more prosaic explanations. Jointly with Paul Milgrom, the famous “multitasking” paper published in JLEO in 1991 notes that contracts shift incentives across different tasks in addition to serving as risk-sharing mechanisms and as methods for inducing effort. Since bonuses on task A will cause agents to shift effort away from hard-to-measure task B, it may be optimal to avoid strong incentives at all (just pay teachers a salary rather than a bonus based only on test performance) or to split job tasks (pay bonuses to teacher A who is told to focus only on math test scores, and pay salary to teacher B who is meant to serve as a mentor). That outcomes are generated by teams also motivates simpler contracts. Holmstrom’s 1982 article on incentives in teams, published in the Bell Journal, points out that if both my effort and yours is required to produce a good outcome, then the marginal product of our efforts are both equal to the entire value of what is produced, hence there is not enough output to pay each of us our marginal product. What can be done? Alchian and Demsetz had noticed this problem in 1972, arguing that firms exist to monitor the effort of individuals working in teams. With perfect knowledge of who does what, you can simply pay the workers a wage sufficient to make the optimal effort, then collect the residual as profit. Holmstrom notes that the monitoring isn’t the important bit: rather, even shareholder controlled firms where shareholders do no monitoring at all are useful. The reason is that shareholders can be residual claimants for profit, and hence there is no need to fully distribute profit to members of the team. Free-riding can therefore be eliminated by simply paying team members a wage of X if the team outcome is optimal, and 0 otherwise. Even a slight bit of shirking by a single agent drops their payment precipitously (which is impossible if all profits generated by the team are shared by the team), so the agents will not shirk. Of course, when there is uncertainty about how team effort transforms into outcomes, this harsh penalty will not work, and hence incentive problems may require team sizes to be smaller than that which is first-best efficient. A third justification for simple contracts is career concerns: agents work hard today to try to signal to the market that they are high quality, and do so even if they are paid a fixed wage. This argument had been made less formally by Fama, but Holmstrom (in a 1982 working paper finally published in 1999 in RESTUD) showed that this concern about the market only completely mitigates moral hazard if outcomes within a firm were fully observable to the market, or the future is not discounted at all, or there is no uncertainty about agent’s abilities. Indeed, career concerns can make effort provision worse; for example, agents may take actions to signal quality to the market which are negative for their current firm! A final explanation for simple contracts comes from Holmstrom’s 1987 paper with Milgrom in Econometrica. They argue that simple “linear” contracts, with a wage and a bonus based linearly on output, are more “robust” methods of solving moral hazard because they are less susceptible to manipulation by agents when the environment is not perfectly known. Michael Powell, a student of Holmstrom’s now at Northwestern, has a great set of PhD notes providing details of these models.

These ideas are reasonably intuitive, but the way Holmstrom answered them is not. Think about how an economist before the 1970s, like Adam Smith in his famous discussion of the inefficiency of sharecropping, might have dealt with these problems. These economists had few tools to deal with asymmetric information, so although economists like George Stigler analyzed the economic value of information, the question of how to elicit information useful to a contract could not be discussed in any systematic way. These economists would have been burdened by the fact that the number of contracts one could write are infinite, so beyond saying that under a contract of type X does not equate marginal cost to marginal revenue, the question of which “second-best” contract is optimal is extraordinarily difficult to answer in the absence of beautiful tricks like the revelation principle partially developed by Holmstrom himself. To develop those tricks, a theory of how individuals would respond to changes in their joint incentives over time was needed; the ideas of Bayesian equilibria and subgame perfection, developed by Harsanyi and Selten, were unknown before the 1960s. The accretion of tools developed by pure theory finally permitted, in the late 1970s and early 1980s, an absolute explosion of developments of great use to understanding the economic world. Consider, for example, the many results in antitrust provided by Nobel winner Jean Tirole, discussed here two years ago.

Holmstrom’s work has provided me with a great deal of understanding of why innovation management looks the way it does. For instance, why would a risk neutral firm not work enough on high-variance moonshot-type R&D projects, a question Holmstrom asks in his 1989 JEBO Agency Costs and Innovation? Four reasons. First, in Holmstrom and Milgrom’s 1987 linear contracts paper, optimal risk sharing leads to more distortion by agents the riskier the project being incentivized, so firms may choose lower expected value projects even if they themselves are risk neutral. Second, firms build reputation in capital markets just as workers do with career concerns, and high variance output projects are more costly in terms of the future value of that reputation when the interest rate on capital is lower (e.g., when firms are large and old). Third, when R&D workers can potentially pursue many different projects, multitasking suggests that workers should be given small and very specific tasks so as to lessen the potential for bonus payments to shift worker effort across projects. Smaller firms with fewer resources may naturally have limits on the types of research a worker could pursue, which surprisingly makes it easier to provide strong incentives for research effort on the remaining possible projects. Fourth, multitasking suggests agent’s tasks should be limited, and that high variance tasks should be assigned to the same agent, which provides a role for decentralizing research into large firms providing incremental, safe research, and small firms performing high-variance research. That many aspects of firm organization depend on the swirl of conflicting incentives the firm and the market provide is a topic Holmstrom has also discussed at length, especially in his beautiful paper “The Firm as an Incentive System”; I shall reserve discussion of that paper for a subsequent post on Oliver Hart.

Two final light notes on Holmstrom. First, he is the source of one of my favorite stories about Paul Samuelson, the greatest economic theorist of all time. Samuelson was known for having a steel trap of a mind. At a light trivia session during a house party for young faculty at MIT, Holmstrom snuck in a question, as a joke, asking for the name of the third President of independent Finland. Samuelson not only knew the name, but apparently was also able to digress on the man’s accomplishments! Second, I mentioned at the beginning of this post the illustrious roster of theorists who once sat at MEDS. Business school students are often very hesitant to deal with formal models, partially because they lack a technical background but also because there is a trend of “dumbing down” in business education whereby many schools (of course, not including my current department at The University of Toronto Rotman!) are more worried about student satisfaction than student learning. With perhaps Stanford GSB as an exception, it is inconceivable that any school today, Northwestern included, would gather such an incredible collection of minds working on abstract topics whose applicability to tangible business questions might lie years in the future. Indeed, I could name a number of so-called “top” business schools who have nobody on their faculty who has made any contribution of note to theory! There is a great opportunity for a Nancy Schwartz or Stan Reiter of today to build a business school whose students will have the ultimate reputation for rigorous analysis of social scientific questions.

Reinhard Selten and the making of modern game theory

Reinhard Selten, it is no exaggeration, is a founding father of two massive branches of modern economics: experiments and industrial organization. He passed away last week after a long and idiosyncratic life. Game theory as developed by the three co-Nobel laureates Selten, Nash, and Harsanyi is so embedded in economic reasoning today that, to a great extent, it has replaced price theory as the core organizing principle of our field. That this would happen was not always so clear, however.

Take a look at some canonical papers before 1980. Arrow’s Possibility Theorem simply assumed true preferences can be elicited; not until Gibbard and Satterthwaite do we answer the question of whether there is even a social choice rule that can elicit those preferences truthfully! Rothschild and Stiglitz’s celebrated 1976 essay on imperfect information in insurance markets defines equilibria in terms of a individual rationality, best responses in the Cournot sense, and free entry. How odd this seems today – surely the natural equilibrium in an insurance market depends on beliefs about the knowledge held by others, and beliefs about those beliefs! Analyses of bargaining before Rubinstein’s 1982 breakthrough nearly always rely on axioms of psychology rather than strategic reasoning. Discussions of predatory pricing until the 1970s, at the very earliest, relied on arguments that we now find unacceptably loose in their treatment of beliefs.

What happened? Why didn’t modern game-theoretic treatment of strategic situations – principally those involve more than one agent but less than an infinite number, although even situations of perfect competition now often are motivated game theoretically – arrive soon after the proofs of von Neumann, Morganstern, and Nash? Why wasn’t the Nash program, of finding justification in self-interested noncooperative reasoning for cooperative or axiom-driven behavior, immediately taken up? The problem was that the core concept of the Nash equilibrium simply permits too great a multiplicity of outcomes, some of which feel natural and others of which are less so. As such, a long search, driven essentially by a small community of mathematicians and economists, attempted to find the “right” refinements of Nash. And a small community it was: I recall Drew Fudenberg telling a story about a harrowing bus ride at an early game theory conference, where a fellow rider mentioned offhand that should they crash, the vast majority of game theorists in the world would be wiped out in one go!

Selten’s most renowned contribution came in the idea of perfection. The concept of subgame perfection was first proposed in a German-language journal in 1965 (making it one of the rare modern economic classics inaccessible to English speakers in the original, alongside Maurice Allais’ 1953 French-language paper in Econometrica which introduces the Allais paradox). Selten’s background up to 1965 is quite unusual. A young man during World War II, raised Protestant but with one Jewish parent, Selten fled Germany to work on farms, and only finished high school at 20 and college at 26. His two interests were mathematics, for which he worked on the then-unusual extensive form game for his doctoral degree, and experimentation, inspired by the small team of young professors at Frankfurt trying to pin down behavior in oligopoly through small lab studies.

In the 1965 paper, on demand inertia (paper is gated), Selten wrote a small game theoretic model to accompany the experiment, but realized there were many equilibria. The term “subgame perfect” was not introduced until 1974, also by Selten, but the idea itself is clear in the ’65 paper. He proposed that attention should focus on equilibria where, after every action, each player continues to act rationally from that point forward; that is, he proposed that in every “subgame”, or every game that could conceivably occur after some actions have been taken, equilibrium actions must remain an equilibrium. Consider predatory pricing: a firm considers lowering price below cost today to deter entry. It is a Nash equilibrium for entrants to believe the price would continue to stay low should they enter, and hence to not enter. But it is not subgame perfect: the entrant should reason that after entering, it is not worthwhile for the incumbent to continue to lose money once the entry has already occurred.

Complicated strings of deductions which rule out some actions based on faraway subgames can seem paradoxical, of course, and did even to Selten. In his famous Chain Store paradox, he considers a firm with stores in many locations choosing whether to price aggressively to deter entry, with one potential entrant in each town choosing one at a time whether to enter. Entrants prefer to enter if pricing is not aggressive, but prefer to remain out otherwise; incumbents prefer to price nonaggressively either if entry occurs or not. Reasoning backward, in the final town we have the simple one-shot predatory pricing case analyzed above, where we saw that entry is the only subgame perfect equilibria. Therefore, the entrant in the second-to-last town knows that the incumbent will not fight entry aggressively in the final town, hence there is no benefit to doing so in the second-to-last town, hence entry occurs again. Reasoning similarly, entry occurs everywhere. But if the incumbent could commit in advance to pricing aggressively in, say, the first 10 towns, it would deter entry in those towns and hence its profits would improve. Such commitment may not possible, but what if the incumbent’s reasoning ability is limited, and it doesn’t completely understand why aggressive pricing in early stages won’t deter the entrant in the 16th town? And what if entrants reason that the incumbent’s reasoning ability is not perfectly rational? Then aggressive pricing to deter entry can occur.

That behavior may not be perfectly rational but rather bounded had been an idea of Selten’s since he read Herbert Simon as a young professor, but in his Nobel Prize biography, he argues that progress on a suitable general theory of bounded rationality has been hard to come by. The closest Selten comes to formalizing the idea is in his paper on trembling hand perfection in 1974, inspired by conversations with John Harsanyi. The problem with subgame perfection had been noted: if an opponent takes an action off the equilibrium path, it is “irrational”, so why should rationality of the opponent be assumed in the subgame that follows? Harsanyi assumes that tiny mistakes can happen, putting even rational players into subgames. Taking the limit as mistakes become infinitesimally rare produces the idea of trembling-hand perfection. The idea of trembles implicitly introduces the idea that players have beliefs at various information sets about what has happened in the game. Kreps and Wilson’s sequential equilibrium recasts trembles as beliefs under uncertainty, and showed that a slight modification of the trembling hand leads to an easier decision-theoretic interpretation of trembles, an easier computation of equilibria, and an outcome that is nearly identical to Selten’s original idea. Sequential equilibria, of course, goes on to become to workhorse solution concept in dynamic economics, a concept which underscores essentially all of modern industrial organization.

That Harsanyi, inventor of the Bayesian game, is credited by Selten for inspiring the trembling hand paper is no surprise. The two had met at a conference in Jerusalem in the mid-1960s, and they’d worked together both on applied projects for the US military, and on pure theory research while Selten visiting Berkeley. A classic 1972 paper of theirs on Nash bargaining with incomplete information (article is gated) begins the field of cooperative games with incomplete information. And this was no minor field: Roger Myerson, in his paper introducing mechanism design under incomplete information – the famous Bayesian revelation principle paper – shows that there exists a unique Selten-Harsanyi bargaining solution under incomplete information which is incentive compatible.

Myerson’s example is amazing. Consider building a bridge which costs $100. Two people will use the bridge. One values the bridge at $90. The other values the bridge at $90 with probability .9, and $30 with probability p=.1, where that valuation is the private knowledge of the second person. Note that in either case, the bridge is worth building. But who should pay? If you propose a 50/50 split, the bridge will simply not be built 10% of the time. If you propose an 80/20 split, where even in their worst case situation each person gets a surplus value of ten dollars, the outcome is unfair to player one 90% of the time (where “unfair” will mean, violates certain principles of fairness that Nash, and later Selten and Harsanyi, set out axiomatically). What of the 53/47 split that gives each party, on average, the same split? Again, this is not “interim incentive compatible”, in that player two will refuse to pay in the case he is the type that values the bridge only at $30. Myerson shows mathematically that both players will agree once they know their private valuations to the following deal, and that the deal satisfies the Selten-Nash fairness axioms: when player 2 claims to value at $90, the payment split is 49.5/50.5 and the bridge is always built, but when player 2 claims to value at $30, the entire cost is paid by player 1 but the bridge is built with only probability .439. Under this split, there are correct incentives for player 2 to always reveal his true willingness to pay. The mechanism means that there is a 5.61 percent chance the bridge isn’t built, but the split of surplus from the bridge nonetheless does better than any other split which satisfies all of Harsanyi and Selten’s fairness axioms.

Selten’s later work is, it appears to me, more scattered. His attempt with Harsanyi to formalize “the” equilibrium refinement, in a 1988 book, was a valiant but in the end misguided attempt. His papers on theoretical biology, inspired by his interest in long walks among the wildflowers, are rather tangential to his economics. And what of his experimental work? To understand Selten’s thinking, read this fascinating dialogue with himself that Selten gave as a Schwartz Lecture at Northwestern MEDS. In this dialogue, he imagines a debate between a Bayesian economist, experimentalist, and an evolutionary biologist. The economist argues that “theory without theorems” is doomed to fail, that Bayesianism is normatively “correct”, and the Bayesian reasoning can easily be extended to include costs of reasoning or reasoning mistakes. The experimentalist argues that ad hoc assumptions are better than incorrect ones: just as human anatomy is complex and cannot be reduced to a few axioms, neither can social behavior. The biologist argues that learning a la Nelson and Winter is descriptively accurate as far as how humans behave, whereas high level reasoning is not. The “chairman”, perhaps representing Selten himself, sums up the argument as saying that experiments which simply contradict Bayesianism are a waste of time, but that human social behavior surely depends on bounded rationality and hence empirical work ought be devoted to constructing a foundation for such a theory (shall we call this the “Selten program”?). And yet, this essay was from 1990, and we seem no closer to having such a theory, nor does it seem to me that behavioral research has fundamentally contradicted most of our core empirical understanding derived from theories with pure rationality. Selten’s program, it seems, remains not only incomplete, but perhaps not even first order; the same cannot be said of his theoretical constructs, as without perfection a great part of modern economics simply could not exist.

Angus Deaton, 2015 Nobel Winner: A Prize for Structural Analysis?

Angus Deaton, the Scottish-born, Cambridge-trained Princeton economist, best known for his careful work on measuring the changes in wellbeing of the world’s poor, has won the 2015 Nobel Prize in economics. His data collection is fairly easy to understand, so I will leave larger discussion of exactly what he has found to the general news media; Deaton’s book “The Great Escape” provides a very nice summary of what he has found as well, and I think a fair reading of his development preferences are that he much prefers the currently en vogue idea of just giving cash to the poor and letting them spend it as they wish.

Essentially, when one carefully measures consumption, health, or generic characteristics of wellbeing, there has been tremendous improvement indeed in the state of the world’s poor. National statistics do not measure these ideas well, because developing countries do not tend to track data at the level of the individual. Indeed, even in the United States, we have only recently begun work on localized measures of the price level and hence the poverty rate. Deaton claims, as in his 2010 AEA Presidential Address (previously discussed briefly on two occasions on AFT), that many of the measures of global inequality and poverty used by the press are fundamentally flawed, largely because of the weak theoretical justification for how they link prices across regions and countries. Careful non-aggregate measures of consumption, health, and wellbeing, like those generated by Deaton, Tony Atkinson, Alwyn Young, Thomas Piketty and Emmanuel Saez, are essential for understanding how human welfare has changed over time and space, and is a deserving rationale for a Nobel.

The surprising thing about Deaton, however, is that despite his great data-collection work and his interest in development, he is famously hostile to the “randomista” trend which proposes that randomized control trials (RCT) or other suitable tools for internally valid causal inference are the best way of learning how to improve the lives of the world’s poor. This mode is most closely associated with the enormously influential J-PAL lab at MIT, and there is no field in economics where you are less likely to see traditional price theoretic ideas than modern studies of development. Deaton is very clear on his opinion: “Randomized controlled trials cannot automatically trump other evidence, they do not occupy any special place in some hierarchy of evidence, nor does it make sense to refer to them as “hard” while other methods are “soft”… [T]he analysis of projects needs to be refocused towards the investigation of potentially generalizable mechanisms that explain why and in what contexts projects can be expected to work.” I would argue that Deaton’s work is much closer to more traditional economic studies of development than to RCTs.

To understand this point of view, we need to go back to Deaton’s earliest work. Among Deaton’s most famous early papers was his well-known development of the Almost Ideal Demand System (AIDS) in 1980 with Muellbauer, a paper chosen as one of the 20 best published in the first 100 years of the AER. It has long been known that individual demand equations which come from utility maximization must satisfy certain properties. For example, a rational consumer’s demand for food should not depend on whether the consumer’s equivalent real salary is paid in American or Canadian dollars. These restrictions turn out to be useful in that if you want to know how demand for various products depend on changes in income, among many other questions, the restrictions of utility theory simplify estimation greatly by reducing the number of free parameters. The problem is in specifying a form for aggregate demand, such as how demand for cars depends on the incomes of all consumers and prices of other goods. It turns out that, in general, aggregate demand generated by utility-maximizing households does not satisfy the same restrictions as individual demand; you can’t simply assume that there is a “representative consumer” with some utility function and demand function equal to each individual agent. What form should we write for aggregate demand, and how congruent is that form with economic theory? Surely an important question if we want to estimate how a shift in taxes on some commodity, or a policy of giving some agricultural input to some farmers, is going to affect demand for output, its price, and hence welfare!

Let q(j)=D(p,c,e) say that the quantity of j consumed, in aggregate is a function of the price of all goods p and the total consumption (or average consumption) c, plus perhaps some random error e. This can be tough to estimate: if D(p,c,e)=Ap+e, where demand is just a linear function of relative prices, then we have a k-by-k matrix to estimate, where k is the number of goods. Worse, that demand function is also imposing an enormous restriction on what individual demand functions, and hence utility functions, look like, in a way that theory does not necessarily support. The AIDS of Deaton and Muellbauer combine the fact that Taylor expansions approximately linearize nonlinear functions and that individual demand can be aggregated even when heterogeneous across individuals if the restrictions of Muellbauer’s PIGLOG papers are satisfied to show a functional form for aggregate demand D which is consistent with aggregated individual rational behavior and which can sometimes be estimated via OLS. They use British data to argue that aggregate demand violates testable assumptions of the model and hence factors like credit constraints or price expectations are fundamental in explaining aggregate consumption.

This exercise brings up a number of first-order questions for a development economist. First, it shows clearly the problem with estimating aggregate demand as a purely linear function of prices and income, as if society were a single consumer. Second, it gives the importance of how we measure the overall price level in figuring out the effects of taxes and other policies. Third, it combines theory and data to convincingly suggest that models which estimate demand solely as a function of current prices and current income are necessarily going to give misleading results, even when demand is allowed to take on very general forms as in the AIDS model. A huge body of research since 1980 has investigated how we can better model demand in order to credibly evaluate demand-affecting policy. All of this is very different from how a certain strand of development economist today might investigate something like a subsidy. Rather than taking obversational data, these economists might look for a random or quasirandom experiment where such a subsidy was introduced, and estimate the “effect” of that subsidy directly on some quantity of interest, without concern for how exactly that subsidy generated the effect.

To see the difference between randomization and more structural approaches like AIDS, consider the following example from Deaton. You are asked to evaluate whether China should invest more in building railway stations if they wish to reduce poverty. Many economists trained in a manner influenced by the randomization movement would say, well, we can’t just regress the existence of a railway on a measure of city-by-city poverty. The existence of a railway station depends on both things we can control for (the population of a given city) and things we can’t control for (subjective belief that a town is “growing” when the railway is plopped there). Let’s find something that is correlated with rail station building but uncorrelated with the random component of how rail station building affects poverty: for instance, a city may lie on a geographically-accepted path between two large cities. If certain assumptions hold, it turns out that a two-stage “instrumental variable” approach can use that “quasi-experiment” to generate the LATE, or local average treatment effect. This effect is the average benefit of a railway station on poverty reduction, at the local margin of cities which are just induced by the instrument to build a railway station. Similar techniques, like difference-in-difference and randomized control trials, under slightly different assumptions can generate credible LATEs. In development work today, it is very common to see a paper where large portions are devoted to showing that the assumptions (often untestable) of a given causal inference model are likely to hold in a given setting, then finally claiming that the treatment effect of X on Y is Z. That LATEs can be identified outside of a purely randomized contexts is incredibly important and valuable, and the economists and statisticians who did the heavy statistical lifting on this so-called Rubin model will absolutely and justly win an Economics Nobel sometime soon.

However, this use of instrumental variables would surely seem strange to the old Cowles Commission folks: Deaton is correct that “econometric analysis has changed its focus over the years, away from the analysis of models derived from theory towards much looser specifications that are statistical representations of program evaluation. With this shift, instrumental variables have moved from being solutions to a well-defined problem of inference to being devices that induce quasi-randomization.” The traditional use of instrumental variables was that after writing down a theoretically justified model of behavior or aggregates, certain parameters – not treatment effects, but parameters of a model – are not identified. For instance, price and quantity transacted are determined by the intersection of aggregate supply and aggregate demand. Knowing, say, that price and quantity was (a,b) today, and is (c,d) tomorrow, does not let me figure out the shape of either the supply or demand curve. If price and quantity both rise, it may be that demand alone has increased pushing the demand curve to the right, or that demand has increased while the supply curve has also shifted to the right a small amount, or many other outcomes. An instrument that increases supply without changing demand, or vice versa, can be used to “identify” the supply and demand curves: an exogenous change in the price of oil will affect the price of gasoline without much of an effect on the demand curve, and hence we can examine price and quantity transacted before and after the oil supply shock to find the slope of supply and demand.

Note the difference between the supply and demand equation and the treatment effects use of instrumental variables. In the former case, we have a well-specified system of supply and demand, based on economic theory. Once the supply and demand curves are estimated, we can then perform all sorts of counterfactual and welfare analysis. In the latter case, we generate a treatment effect (really, a LATE), but we do not really know why we got the treatment effect we got. Are rail stations useful because they reduce price variance across cities, because they allow for increasing returns to scale in industry to be utilized, or some other reason? Once we know the “why”, we can ask questions like, is there a cheaper way to generate the same benefit? Is heterogeneity in the benefit important? Ought I expect the results from my quasiexperiment in place A and time B to still operate in place C and time D (a famous example being the drug Opren, which was very successful in RCTs but turned out to be particularly deadly when used widely by the elderly)? Worse, the whole idea of LATE is backwards. We traditionally choose a parameter of interest, which may or may not be a treatment effect, and then choose an estimation technique that can credible estimate that parameter. Quasirandom techniques instead start by specifying the estimation technique and then hunt for a quasirandom setting, or randomize appropriately by “dosing” some subjects and not others, in order to fit the assumptions necessary to generate a LATE. If is often the case that even policymakers do not care principally about the LATE, but rather they care about some measure of welfare impact which rarely is immediately interpretable even if the LATE is credibly known!

Given these problems, why are random and quasirandom techniques so heavily endorsed by the dominant branch of development? Again, let’s turn to Deaton: “There has also been frustration with the World Bank’s apparent failure to learn from its own projects, and its inability to provide a convincing argument that its past activities have enhanced economic growth and poverty reduction. Past development practice is seen as a succession of fads, with one supposed magic bullet replacing another—from planning to infrastructure to human capital to structural adjustment to health and social capital to the environment and back to infrastructure—a process that seems not to be guided by progressive learning.” This is to say, the conditions necessary to estimate theoretical models are so stringent that development economists have been writing noncredible models, estimating them, generating some fad of programs that is used in development for a few years until it turns out not to be silver bullet, then abandoning the fad for some new technique. Better, the randomistas argue, to forget about external validity for now, and instead just evaluate the LATEs on a program-by-program basis, iterating what types of programs we evaluate until we have a suitable list of interventions that we feel confident work. That is, development should operate like medicine.

We have something of an impasse here. Everyone agrees that on many questions theory is ambiguous in the absence of particular types of data, hence more and better data collection is important. Everyone agrees that many parameters of interest for policymaking require certain assumptions, some more justifiable than others. Deaton’s position is that the parameters of interest to economists by and large are not LATEs, and cannot be generated in a straightforward way from LATEs. Thus, following Nancy Cartwright’s delightful phrasing, if we are to “use” causes rather than just “hunt” for what they are, we have no choice but to specify the minimal economic model which is able to generate the parameters we care about from the data. Glen Weyl’s attempt to rehabilitate price theory and Raj Chetty’s sufficient statistics approach are both attempts to combine the credibility of random and quasirandom inference with the benefits of external validity and counterfactual analysis that model-based structural designs permit.

One way to read Deaton’s prize, then, is as an award for the idea that effective development requires theory if we even hope to compare welfare across space and time or to understand why policies like infrastructure improvements matter for welfare and hence whether their beneficial effects will remain when moved to a new context. It is a prize which argues against the idea that all theory does is propose hypotheses. For Deaton, going all the way back to his work with AIDS, theory serves three roles: proposing hypotheses, suggesting which data is worthwhile to collect, and permitting inference on the basis of that data. A secondary implication, very clear in Deaton’s writing, is that even though the “great escape” from poverty and want is real and continuing, that escape is almost entirely driven by effects which are unrelated to aid and which are uninfluenced by the type of small bore, partial equilibrium policies for which randomization is generally suitable. And, indeed, the best development economists very much understand this point. The problem is that the media, and less technically capable young economists, still hold the mistaken belief that they can infer everything they want to infer about “what works” solely using the “scientific” methods of random- and quasirandomization. For Deaton, results that are easy to understand and communicate, like the “dollar-a-day” poverty standard or an average treatment effect, are less virtuous than results which carefully situate numbers in the role most amenable to answering an exact policy question.

Let me leave you three side notes and some links to Deaton’s work. First, I can’t help but laugh at Deaton’s description of his early career in one of his famous “Notes from America”. Deaton, despite being a student of the 1984 Nobel laureate Richard Stone, graduated from Cambridge essentially unaware of how one ought publish in the big “American” journals like Econometrica and the AER. Cambridge had gone from being the absolute center of economic thought to something of a disconnected backwater, and Deaton, despite writing a paper that would win a prize as one of the best papers in Econometrica published in the late 1970s, had essentially no understanding of the norms of publishing in such a journal! When the history of modern economics is written, the rise of a handful of European programs and their role in reintegrating economics on both sides of the Atlantic will be fundamental. Second, Deaton’s prize should be seen as something of a callback to the ’84 prize to Stone and ’77 prize to Meade, two of the least known Nobel laureates. I don’t think it is an exaggeration to say that the majority of new PhDs from even the very best programs will have no idea who those two men are, or what they did. But as Deaton mentions, Stone in particular was one of the early “structural modelers” in that he was interested in estimating the so-called “deep” or behavioral parameters of economic models in a way that is absolutely universal today, as well as being a pioneer in the creation and collection of novel economic statistics whose value was proposed on the basis of economic theory. Quite a modern research program! Third, of the 19 papers in the AER “Top 20 of all time” whose authors were alive during the era of the economics Nobel, 14 have had at least one author win the prize. Should this be a cause for hope for the living outliers, Anne Krueger, Harold Demsetz, Stephen Ross, John Harris, Michael Todaro and Dale Jorgensen?

For those interested in Deaton’s work beyond what this short essay, his methodological essay, quoted often in this post, is here. The Nobel Prize technical summary, always a great and well-written read, can be found here.

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.

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