Category Archives: Classics

“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.”)

“A Penny for your Quotes: Patent Citations and the Value of Innovation,” M. Trajtenberg (1990)

This is one of those classic papers where the result is so well-known I’d never bothered to actually look through the paper itself. Manuel Trajtenberg, in the late 1980s, wrote a great book about Computed Tomography, or CAT scans. He gathered exhaustive data on sales by year to hospitals across the US, the products/attributes available at any time, and the prices paid. Using some older results from economic theory, a discrete choice model can be applied to infer willingness-to-pay for various types of CAT scanners over time, and from there to infer the total social surplus being generated at any time. Even better, Trajtenberg was able to calculate the lifetime discounted value of innovations occurring during any given period by looking at the eventual diffusion path of those technologies; that is, if a representative consumer is willing to pay Y in 1981 for CAT scanner C, and the CAT scanner diffused to 50 percent market share over the next five years, we can integrate the willingness to pay over the diffusion curve to get a rough estimate of the social surplus generated. CAT innovations during their heyday (roughly the 1970s, before MRI began to diffuse) generated about 17 billion dollars of surplus in 1982 dollars.

That alone is interesting, but Trajtenberg takes this fact one step further. There has long been a debate about whether patent citations tell you much about actual innovation. We know from a variety of sources that most important inventions are not patented, that many low-quality inventions of little social value are patented, and that patents are used in enormously different ways depending on market structure. Since Trajtenberg has an actual measure of social welfare created by newly-introduced products in each period, and a measure of industry R&D in each period, and a measure counting patents issued in CT in each period (nearly 500 in total), he can actually check: is patenting activity actually correlated with socially beneficial innovation?

The answer, it turns out, is no. A count of patents, at any reasonable lag and any restriction to “core” CT firms or otherwise, never has a correlation with change in total social value of more than .13. On the other hand, patents lagged five months has a correlation of .933 with industry R&D. No surprise, R&D appears to buy patents at a pretty constant rate, but not to buy important breakthroughs. This doesn’t, however, mean patent data is worthless to the analyst. Instead of looking at patents, we can look at citation-weighted patents. A patent that gets cited 10 times is surely more important than one which is issued and never heard from again. Weighing patents by citation count, the correlation between the number of weighted patents (lagged a few months to give products time to reach the market) and total social welfare created is in the area of .75! This result has been confirmed many, many, many times since Trajtenberg’s paper. Harhoff et al (1999) found, using survey data, that each single patent citation for highly-cited patents is a signal that the patent has a additional private value of a million US dollars. Hall, Jaffe and Trajtenberg (2005) found that, using Tobin’s Q on stock market data holding firm R&D and total number of patents constant, an additional patent citation improves firm value by an average of 3%.

Final 1990 RAND copy (IDEAS page).

“Price Formation of Fish,” A.P Barten & L.J. Bettendorf (1989)

I came across this nice piece of IO in a recent methodological book by John Sutton, which I hope to cover soon. Sutton recalls Lionel Robbins’ famous Essay on the Nature of Significance of Economic Science. In that essay, Robbins claims the goal of the empirically-minded economist is to estimate stable (what we now call “structural”) parameters whose stability we know a priori from theory. (As an aside, it is tragic that Hurwicz’ 1962 “On the Structural Form of Interdependent Systems”, from which Robbins’ idea gets its modern treatment, is not freely available online; all I see is a snippet from the conference volume it appeared at here). Robbins gives the example of an empiricist trying to estimate the demand for haddock by measuring prices and quantities each day, controlling for weather and the like, and claiming that the average elasticity has some long-run meaning; this, he says, is a fool’s errand.

Sutton points out how interesting that example is: if anything, fish are an easy good to examine! They are a good with easy-to-define technical characteristics sold in competitive wholesale markets. Barten and Bettendorf point out another interesting property: fish are best described by an inverse demand system, where consumers determine the price paid as a function of the quantity of fish in the market rather than vice versa, since quantity in the short run is essentially fixed. To the theorist, there is no difference between demand and inverse demand, but to the empiricist, that little error term must be added to the exogenous variables if we are to handle statistical variation correctly. Any IO economist worth their salt knows how to estimate common demand systems like AIDS, but how should we interpret parameters in inverse demand systems?

Recall that, in theory, Marshallian demand is a homogeneous of degree zero function of total expenditures and prices. Using the homogeneity, we have that the vector quantity demand q is a function of P, the fraction of total expenditure paid for each unit of each good. Inverting that function gives P as a function of q. Since inverse demand is the result of a first-order condition from utility maximization, we can restate P as a function of marginal utilities and quantities. Taking the derivative of P, with some judicious algebra, one can state the (normalized) inverse demand as the sum of moves along an indifference surface and moves across indifference surfaces; in particular, dP=gP’dq+Gdq, where g is a scalar and G is an analogue of the Slutsky matrix for inverse demand, symmetric and negative semidefinite. All we need to do know is to difference our data and estimate that system (although the authors do a bit more judicious algebra to simplify the computational estimation).

One more subtle step is required. When we estimate an inverse demand system, we may wish to know how substitutable or complementary any two goods are. Further, we want such an estimate to be invariant to arbitrary monotone increasing changes in an underlying utility function (the form of which is not assumed here). It turns out that Allais (in his 1943 text on “pure economics” which, as far as I know, is yet to be translated!) has shown how to construct just such a measure. Yet another win for theory, and for Robbins’ intuition: it is hopeless to atheoretically estimate cross-price elasticities or similar measures of substitutability atheoretically, since these parameters are determined simultaneously. It is only as a result of theory (here, nothing more than “demand comes from utility maximizers” is used) that we can even hope to tease out underlying parameters like these elasticities. The huge numbers of “reduced-form” economists these days who do not understand what the problem is here really need to read through papers of this type; atheoretical training is, in my view, a serious danger to the grand progress made by economics since Haavelmo and Samuelson.

It is the methodology that is important here; the actual estimates are secondary. But let’s state them anyway: the fish sold in the Belgian markets are quite own-price elastic, have elasticities that are consistent with demand-maximizing consumers, and have patterns of cross-price elasticities across fish varieties that are qualitatively reasonable (bottom-feeders are highly substitutable with each other, etc.) and fairly constant across a period of two decades.

Final version in EER (No IDEAS version). This paper was in the European Economic Review, an Elsevier journal that is quickly being killed off since the European Economic Association pulled out of their association with Elsevier to run their own journal, the JEEA. The editors of the main journal in environmental economics have recently made the same type of switch, and of course, a group of eminent theorist made a similar exit when Theoretical Economics began. Jeff Ely has recently described how TE came about; that example makes it quite clear that journals are actually quite inexpensive to run. Even though we economists are lucky to have nearly 100% “green” open access, where preprints are self-archived by authors, we still have lots of work to do to get to a properly ungated world. The Econometric Society, for example, spends about $900,000 for all of its activities aside from physically printing journals, a cost that could still be recouped in an open access world. Much of that is for running conferences, giving honoraria, etc, but let us be very conservative and estimate no income is received aside from subscriptions to its three journals, including archives. This suggests that a complete open access journal and archives for the 50 most important journals in the field requires, very conservatively, revenue of $15 million per year, and probably much less. This seems a much more effective use of NSF and EU moneys that funding a few more graduate research assistants.

“Without Consent or Contract,” R. W. Fogel (1989)

Word comes that Bob Fogel, an absolute giant in economic history and a Nobel Prize winner, passed away today. I first encountered Fogel in a class a decade or so ago taught by Robert Margo, another legendary scholar of the economics of American history.

Fogel’s most famous contribution is summarized in the foreword to the very readable Without Consent or Contract. “Although the slave system was horribly retrogressive in its social, political, and ideological aspects, it was quite advanced by the standards of the time in its technology and economic organization. The paradox is only apparent…because the paradox rests on the widely held assumption that technological efficiency is inherently good. It is this beguiling assumption that is false and, when applied to slavery, insidious.”

Roughly, it was political change alone, not economic change, which could have led to the end of slavery in America. The plantation system was, in fact, a fairly efficient system in the economic sense, and was not in danger of petering out on its own accord. Evidence on this point was laid out in technical detail in Fogel and Engerman’s “Time on the Cross”. In that text, evidence from an enormous number of sources is brought to bear on the value of a slave over time; McCloskey has called Fogel “a carpenter of history…measure, measure again, measure again.” The idea that the economic effects of history can be (and are) wildly different from the moral or political effects remains misunderstood; Melissa Dell’s wonderful paper on the Peruvian mita is a great example of a terrible social policy which nonetheless had positive long-run economic effects. As historians disdain “Whig history”, the idea that things improve as time marches on, economists ought disdain “Whig economics”, the idea that growth-inducing policies are somehow linked to moral ones.

There is much beyond the slavery research, of course. In one of the most famous papers in economic history, Fogel studied the contribution of the railroad to American economic growth (Google has this at only 86 citations; how is such a low number possible?). He notes that, as economists, we should care about the marginal benefit, not the absolute benefit, of the railroad. In the absence of rail, steamboats and canals were still possible (and would likely have been built in the midwest). He famously claims that the US would have reached its income in January 1890 by the end of March 1890 had there been no rail at all, a statement very much contrary to traditional historical thinking.

Fogel’s later life was largely devoted to his project on the importance of improved nutrition and its interaction with economic growth, particularly since the 1700s. If you’ve not seen these statistics, it is amazing just how short and skinny the average human was before the modern era. There has been an enormous debate over the relative role of nutrition, vis-a-vis technologies, knowledge like germ theory, or embodied or diffused knowledge, in the increased stature of man: Angus Deaton summarizes the literature nicely. In particular, my read is that the thesis whereby better nutrition causes a great rise in human incomes is on fairly shaky ground, though the debate is by no means settled.

Amazon has Without Consent or Contract for sale for under 15 bucks, well worth it. Some quick notes: Fogel was by no means a lone voice in cliometrics; for example, Conrad and Meyer in a 1958 JPE make very much the same point as Fogel concerning the economic success of slavery, using tools from capital theory in the argument. Concerning the railroad, modern work suggests Fogel may have understated its importance. Donaldson and Hornbeck, two of the best young economic historians in the world, use some developments in modern trade theory to argue that increased market access due to rail, measured as market access is capitalized into farmland, was far more important to GDP growth than Fogel suggested.

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.

“Returns to Scale in Research & Development: What Does the Schumpeterian Hypothesis Imply?,” F. Fisher & P. Temin (1973)

Schumpeter famously argued for the economic importance of market power. Even though large firms cause static inefficiency, they had dynamic benefits in that large firms demand more invention since they can extract more revenue from each new product. Further, they supply more invention, Schumpeter hypothesized, since the rate of invention has increasing returns to scale in the number of inventors, and in the number of other employees at the firm. (Axioms A and B). The second part of that statement may be for many reasons; for instance, if the output of a research project could be many potential products, a larger firm has the ability to capitalize on many of those new projects, whereas a small firm might have more limited complementary capabilities. Often, this hypothesis has been tested by checking whether larger firms are more research intensive, meaning that larger firms have a higher percentage of their workforce doing research (Hypothesis 1). Alternatively, a direct reading of Schumpeter is that a 1% increase in the non-research staff of a firm leads to a more than 1% increase in total R&D output of a firm, where output is just the number of research workers times each worker’s average output as a function of firm size (Hypothesis 2).

And here is where theory comes into play. Are axioms A and B necessary or sufficient for either hypothesis 1 or 2? If they don’t imply hypothesis 1, then the idea of testing the Schumpeterian axioms about increasing returns to scale by examining researcher employment is wrong-headed. If they don’t imply hypothesis 2, then Schumpeter’s qualitative argument is incomplete in the first place. Fisher and Temin (that’s Franklin Fisher and Peter Temin, two guys who, it goes without saying, have had quite some careers since they wrote this paper in the early 70s!) show that, in fact, for both hypotheses the axioms are neither necessary nor sufficient.

An even more basic problem wasn’t noticed by Fisher and Temin, but instead was pointed out by Carlos Rodriguez in a 1979 comment. If Axiom 1 holds, and the average product per researcher is increasing in the number of researchers, then marginal product always exceeds average product. If market equilibrium means I pay all research workers their marginal product, then I will be making a loss if I operate at the “optimal” quantity. Hence I will hire no research workers at all. So step one to interpreting Schumpeter, then, is to restate his two axioms. A weaker condition might be that if the number of research and the number of nonresearch workers increase at the same rate, then average product per research worker is increasing. This is implied by Axioms A and B, but doesn’t rely on always-increasing average product per research worker (Axiom C). This is good for checking our two hypotheses, since anything that would have been implied by Axioms A and B is still implied by our more theoretically-grounded axiom C.

So what does our axiom imply about the link between research staff size and firm size? Unsurprisingly, nothing at all! Surely the optimal quantity of research workers depends on the marginal product of more research workers as firm size grows, and not on the average product of those workers. Let’s prove it. Let F(R,S) is the average product per research worker as a function of R, the number of researchers, and S, the number of other employees at the firm. I hire research workers as long as their marginal product exceeds the researcher wage rate. The marginal product of total research output is the derivative of R*F(R,S) with respect to R, or F+R*dF/dR. As S increases, this marginal product goes up if and only if dF/dS+R*dF^2/dRdS>0. That is, I hire more research workers in equilibrium if my non-research staff is bigger according to a function that depends on the second derivative of the average output per researcher. But my axioms had only to do with the first derivative! Further, if dF/dS+R*dF^2/dRdS>0, then larger firms have a larger absolute number of scientists than smaller firms, but this implication is completely independent of the Schumpeterian axioms. What’s worse, even that stronger assumption involving the second derivative does not imply anything about the share of research workers on the staff.

The moral is the same one you were probably taught you first day of economics class: using reasoning about averages to talk about equilibrium behavior, so dependent on marginals, can lead you astray very quickly!

1971 working paper; the final version was published in JPE 1973 (IDEAS). Related to the comment by Rodriguez, Fisher and Temin point out here that the problem with increasing returns to scale does not ruin their general intuition, for the reasons I stated above. What about the empirics of Schumpeter’s prediction? Broadly, there is not much support for a link between firm size and research intensity, though the literature on this is quite contentious. Perhaps I will cover it in another post.

“The Meaning of Utility Measurement,” A. Alchian (1953)

Armen Alchian, one of the dons from UCLA’s glory days, passed away today at 98. His is, for me, a difficult legacy to interpret. On the one hand, Alchian-Demsetz 1972 is among the most famous economics papers ever written, and it can fairly be considered the precursor to mechanism design, the most important new idea in economics in the past 50 years. People produce more by working together. It is difficult to know who shirks when we work as a team. A firm gives a residual claimant (an owner) who then has an incentive to monitor shirking, and as only one person needs to monitor the shirking, this is much less costly than a market where each member of the team production would need somehow to monitor whether other parts of the team shirk. Firms are deluded if they think that they can order their labor inputs to do whatever they want – agency problems exist both within and outside the firm. Such an agency theory of the firm is very modern indeed. That said, surely this can’t explain things like horizontally integrated firms, with different divisions producing wholly different products (or, really, any firm behavior where output is a separable function of each input in the firm).

Alchian’s other super famous work is his 1950 paper on evolution and the firm. As Friedman would later argue, Alchian suggested that we are justified treating firms as if they are profit maximizers when we do our analyses since the nature of competition means that non-profit maximizing firms will disappear in the long run. I am a Nelson/Winter fan, so of course I like the second half of the argument, but if I want to suggest that firms partially seek opportunities and partially are driven out by selection (one bit Lamarck, one bit Darwin), then why not just drop the profit maximization axiom altogether and try to write a parsimonious description of firm behavior which doesn’t rely on such maximization?

It turns out that if you do the math, profit maximization is not generally equivalent to selection. Using an example from Sandroni 2000, take two firms. There are two equally likely states of nature, Good and Bad. There are two things a firm can do, the risky one, which returns profit 3 in good states and 0 in bad states, and a risk-free one, which always returns 1. Maximizing expected profit means always investing all capital in the risky state, hence eventually going bankrupt. A firm who doesn’t profit maximize (say, it has incorrect beliefs and thinks we are always in the Bad state, hence always takes the risk-free action) can survive. This example is far too simple to be of much worth, but it does at least remind us of lesson in the St. Petersburg paradox: expected value maximization and survival have very little to do with each other.

More interesting is the case with random profits, as in Radner and Dutta 2003. Firms invest their capital stock, choosing some mean-variance profits pair as a function of capital stock. The owner can, instead of reinvesting profits into the capital stock, pay out to herself or investors. If the marginal utility of a dollar of capital stock falls below a dollar, the profit-maximizing owner will not reinvest that money. But a run of (random) losses can drive the firm to bankruptcy, and does so eventually with certainty. A non-profit maximizing firm may just take the lowest variance earnings in every period, pay out to investors a fraction of the capital stock exactly equal to the minimum earnings that period, and hence live forever. But why would investors ever invest in such a firm? If investment demand is bounded, for example, and there are many non profit-maximizing firms from the start, it is not the highest rate of return but the marginal rate of return which determines the market interest rate paid to investors. A non profit-maximizer that can pay out to investors at least that much will survive, and all the profit maximizers will eventually fail.

The paper in the title of this post is much simpler: it is merely a very readable description of von Neumann expected utility, when utility can be associated with a number and when it cannot, and the possibility of interpersonal utility comparison. Alchian, it is said, was a very good teacher, and from this article, I believe it. What’s great is the timing: 1953. That’s one year before Savage’s theory, the most beautiful in all of economics. Given that Alchian was associated with RAND, where Savage was fairly often, I imagine he must have known at least some of the rudiments of Savage’s subjective theory, though nothing appears in this particular article. 1953 is also two years before Herbert Simon’s behavioral theory. When describing the vN-M axioms, Alchian gives situations which might contradict each, except for the first, a complete and transitive order over bundles of goods, an assumption which is consistent with all but “totally unreasonable behavior”!

1953 AER final version (No IDEAS version).

“The Oligopoly Solution Concept is Identified,” T. Bresnahan (1980)

Here’s a classic, super-simple paper. I think I can give you the idea in two paragraphs. I know price and quantity sold for some good. I know supply and demand must equate. I use whatever method I like to deal with simultaneity of the supply and demand functions (say, a cost shifter approach). How can I identify whether the industry is acting as if it had market power? That is, how can I separate collusive behavior from competitive behavior?

A numerical example will help. Let marginal costs and demand be linear. Let demand be P=11-Q. Shift demand and supply (meaning shift the intercept) however you like. The price-quantity bundle you see under monopoly with MC constant and equal to 1 will be identical to the price-quantity bundle you see under perfect competition with MC increasing and equal to 1+Q. For instance, price=6 and quantity=5 is found by letting P=MC for MC=1+Q or by letting MR=MC for MR=11-2Q. And demand shifters don’t help us! If demand shifts to P=R-Q, where R is any y-intercept, then under perfect competition with MC=1+Q, we have equilibrium price such that 1+Q=R-Q, or Q=(R-1)/2, and equilibrium price under monopoly with MC=1 such that MC=MR, or R-2Q=1, or Q=(R-1)/2. Supply shifters are equally unhelpful: for inverse demand P=11-Q, shifting the y-intercept for the cost curve of both the hypothetical monopolist and the hypothetical perfectly competitive market changes equilibrium quantity by exactly the same amount. So what to do? The simplest method is to assume, a priori, something about the nature of marginal costs in the industry; if they are constant, the the price patterns we saw in the numerical example can only be explained by monopoly/collusive behavior. But Bresnahan points out that we don’t even need to make this assumption. Just note that a rotation of the demand curve through some equilibrium point affects those with market power and those without differently. Since rotating the demand curve retains the P=MC equilibrium condition under perfect competition, such a rotation only affects equilibrium price and quantity if competition is not perfect. If I have, say, demand-side instruments, one of which only affects the y-intercept and one of which affects the slope (and perhaps also the intercept), then not only can I identify whether perfect competition exists, but I can even identify the degree to which behavior is monopolistic. Useful.

Final version from Economic Letters 1982 (IDEAS)

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

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

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

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

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

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

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