Broken Symmetry

Thouhts on Nick Paumgarten's "The Secret Cycle"

Nick Paumgarten's piece, "The Secret Cycle" in the 12 October 2009 New Yorker is probably worth a read for some readers of Broken Symmetry. The article, which is unfortunately gated, retells some of the history of technical forecasting based on observed periodicities in a variety of independent variables, including (naturally) market prices.

Armstrong sounds like an idiot savant to me. He probably is quite good at the technical analysis, but is full of @#$# when it comes to his speculations about what mechanisms might account for observed periodicities. So are most people who claim to see signal in the noise of market prices. For the record, your author endeavors to invest only with a margin of safety calculated Graham-and-Dodd-style (or, better, Buffett and Munger-style) from fundamentals.

But long-time readers of Broken Symmetry may recall earlier posts on a "fundamental hypothesis of periodicity." Also for the record, I will say that I believe there are probably collective modes of oscillation produced by organizations of individuals that arise from weak nonlinear couplings of human behavior. That belief is not pure speculation, as the papers from Malmgren and Watts and on synchronized clusters demonstrate.

I am profoundly skeptical -- in fact, I believe it's impossible -- for us to accurately forecast market price cycles out for years into the future with day-level precision as people like Martin Armstrong claim to be able to do. Yet I believe it's not impossible that much more of our social and economic lives are governed by low frequency, but large-scale and low-magnitude "forces" than most economists would admit. It's just that the higher energy noise of everyday life makes the influence of these weak forces much harder to discern. It's only in moments of profound social uncertainty -- such as the present crisis -- that people begin to wonder how much of their individual destiny is truly independent of their neighbors'.

Robert Musil says somewhere that we are as connected to each other as two drops in a river. That has always sounded about right to me. And if you're thinking, "That's not too connected!" then I refer you to consider the chemistry of water more carefully (and particularly the hydrogen bond), and remind you that Musil was trained in that chemistry.

What Mandelbrot and Fama Both Got Wrong: Buyers and Sellers Can Synchronize

I've been working through a paper by Eugene Fama, titled "Madelbrot and the Stable Paretian Hypothesis." The paper is from 1963. I found it in Modern Developments in Investment Management, edited by Lorie and Brealey and published in 1976. It was in the free books bin outside a bookstore in Noe Valley!

As a historical record, this is a wonderful article. In the article, Fama finds the data inconclusive as to whether the Pareto distribution applies better as a fit to the distribution of price changes during a given period of time. The punchline, of course, is that Fama's efficient market hypothesis (which he developed in the decade after this paper was published) gave the confidence at least to Black and Scholes and many others to base their models of price changes on Gaussian distributions. At least at one point, Fama was open to another possibility.

But what is more interesting is the evidence from this paper, which suggests that both Mandelbrot and Fama may have made the same mistake in analyzing time series of prices. In particular, it seems that both Mandelbrot and Fama wanted to maintain the assumption of independence in price changes in successive intervals. They seem to have had the same reason too. Both wanted whatever distribution that might apply in a shorter interval to apply also in longer or successive intervals -- the very condition that fails when the price changes are not independent. Here's how Fama puts it:

If the changes between transactions are independent, identically distributed, stable Paretian variables, daily, weekly, and monthly changes will follow stable Paretian distributions of exactly the same form, except for origin and scale.

Mandelbrot, recognizing that a Gaussian was not a good fit, picked the Pareto distribution as an alternative. Why did he choose the Pareto distribution? I'm not sure, but there is a good chance that he did so because while its infinite variance permitted wider swings in price than the Gaussian, the Pareto distribution is still stationary in the sense that it treats the price changes during particular intervals as wholly independent.

Both Mandelbrot and Fama seem to have let themselves be guided too strongly by their theoretical ambitions. If we observe what actually happens in crowded marketplaces (such as a crowded trading floor), it is easy to observe how buying and selling behavior is correlated. This kind of correlation (which Keynes attributed to "animal spirits") destroys the very independence that is necessary to the assumption of stationarity made by both Mandelbrot and Fama in their models of market prices!

What alternative models might we use to understand price changes? We have to start from models like the 1D Ising model described in the last post, models which show how correlations at the microeconomic scale can aggregate up into macroeconomic fluctuations. We must begin with the assumption that buying and selling decisions are at least weakly coupled, so that market price changes reflect the large swings that can result from synchronization. In future posts, I shall attempt an explanation for how the Kuramoto Model of synchronization could be used to model the non-stationary distribution(s) of market prices.

UPDATE: My sincere apologies to Mandelbrot. I find in this early work that he did not assume stationarity, and was aware of the non-stationarity in these distributions all along.

Mario Rizzo's Reply to Robert Lucas

From a new favorite, the ThinkMarkets blog:

what Hayek makes clear is that it would not be too much to ask of a theory that it at least point in the general direction of a problem. Lucas, and others, continue to think of recent events as purely a bubble phenomenon — and so there is a degree of randomness involved in the phenonemon, in their view. However, Hayek’s cycle theory pointed to an interest rate effects on higher-order investments that generally or systematically produce over- and mal-investment. But it says little about timing. Lucas seems oblivious to both.

Hayek's vision of the macroeconomy is at least consistent with an explanation of bubbles as a self-organized critical (SOC) phenomenon. For an introduction for those with background in physical sciences or mathematics, see the short treatise by Jensen. What these phenomena have in common is a large system of N components (such as atoms, molecules, sand grains, trees in a forest) that are driven into more-ordered states by a process that acts slowly over time (such as a low-frequency magnetic force, temperature gradient, or growth of a forest) and into less-ordered states by a process that happens rapidly through a mechanism of dense interconnections between components (such as electron-electron interactions, or the spreading of a forest fire). Not surprisingly since we now know that no system is deterministic, SOC does not let you predict when the system will relax. What it does let you do is predict what statistics will apply to the dynamical processes, and how those statistics are affected by the microdynamical variables.

In the case of a macroeconomy, SOC theory would suggest that an increase in money supply that is slow in comparison to the rate of exchange among market actors would generally lead to an overall increase in wealth over some period of time followed by rapid relaxations (i.e., crashes) once the patterns of activity encouraged by cooperation and exchange within a period exhaust their social benefits.

How Economics Can Become Scientific

The MIT Technology Review blog today reviews a "paper describ[ing] a survey taken by graduate students at seven top-ranking economics programs" on answers to questions, including the following:

• Is neoclassical economics relevant?
• Do economists agree on fundamental issues?
• Can fiscal policy be an effective stabilizer?
• Should the Fed maintain a constant growth rate of the money supply?

The results are mixed. Not a surprise since even among economists the answers to these questions are mixed! But the title of the post raises a large question, which I'd like to answer using systems theory -- i.e., the theory of how highly integrated social networks self-organize. Systems theory has been a central theme for this blog.

If we understand the question in terms of systems, then we can answer in layers the question, "Is economics a science?" These categories are linked to the three generalized components that make "the system of economics":

1. The domain of rules and procedures that make up economic theory;
2. the field of recognized experts who act as gatekeepers to the domain; and
3. the stream of data in the form of observations or measurements used to falsify the domain.

The high-level answer is that economics is a science if it has active components of domain, field, and data. Economics is not only dismal, but dead if none of these is changing. But so long as each of these three have some time-dependence, the difference between economics and other sciences is a difference in degree, not kind.

But on closer inspection, there are indeed apparent problems with these components for economics that do not exist for other sciences. The biggest problem seems to be that at least until about ten or twenty years ago, there was no stream of data on individual or group behavior that could be used to falsify the domain of neoclassical theory that developed around WWII. In other words, there have been few new experiments or new kinds of data sets since neoclassical theory was developed. The field has become somewhat retrenched as it has labored at refining its theories using the same sets of data over and over again.

There are exceptions, of course. I'm in the middle of reading Vernon Smith's Rationality in Economics, in which he describes the interesting ways that actual behavior deviates from neoclassical predictions in actual experiments and field studies. And Smith is not laboring in obscurity, of course. He won a Nobel for his efforts in this regard. But in general, the "science" of economics seems to have been limited by a lack of data. There simply has not been an experimental or observational mechanism for the field to reach consensus on what hypotheses have been falsified.

In perhaps not-so-humble fashion for a non-economist, I would like to propose that experiments and observations of the average time period between individual acts of consumption or production is exactly the kind of data that economists could use to falsify models of individual or group behavior. The fact that such observations have not been made in the past is no accident, for they were generally too expensive to be collected before the Internet became widely available. (Csikszentmihalyi's Experience Sampling Method is a notable exception.)

Journal entries themselves encode the information that is necessary for managers, investors, and owners of corporations to see the frequency spectrum of activity within a firm that provides a unique signature of the firm's activities. Time-averaged measures of costs are not the only variables important to the health of a corporation. Profit is a function of sales and costs of sales over time.

Economics could become more scientific by statistical observations of individual and group behavior over time.

Why I love Paul Graham

Check out his latest essay here.

But there's another way of using time that's common among people who make things, like programmers and writers. They generally prefer to use time in units of half a day at least. You can't write or program well in units of an hour. That's barely enough time to get started.

When you're operating on the maker's schedule, meetings are a disaster. A single meeting can blow a whole afternoon, by breaking it into two pieces each too small to do anything hard in. Plus you have to remember to go to the meeting.

The humor and sincerity of his essays is rare.

Weighing in on the EMH Debate: Does Behavioral Analysis Resolve the Problems?

See comments from the Economist, DeLong, Yglesias, and Hansen.

The problem with EMH is with its interpretation of the principle of revealed preferences. Unfortunately, the same problematic interpretation often applies to behavioral critiques of EMH. Here's an excerpt from an article I have forthcoming in Intellectual Asset Management:

There are two catches to using preferences "revealed" by market prices to measure individual preferences. The first catch is that one has to watch over a period of time to see the preferences revealed, which means assuming that the preferences do not change during the period of observation. The second catch is that in practice, no large data sets have recorded the revealed preferences of individuals. Only aggregate data, such as consumer price indexes and average household spending over several decades, has been available. Taken together, these catches mean that empirical tests of the rational hypothesis take as a given that preferences do not change in time and that only information about the average preferences of a group is necessary to measure and predict individual preferences.

But preferences do change over time. New books are written, new paintings are painted, new inventions are conceived and reduced to practice. Moreover, no actual person may exist who embodies the average preferences of a large group. Especially if there are outliers, the average preferences of the group may be quite different from the preferences of most individuals within the group.

Unfortunately, many advocates of “behavioral economics” and “bounded rationality” seem to have made the same mistake as the EMH by interpreting the rational hypothesis as a hypothesis about individual behavior. Although it is true that individual behavior often deviates predictably from what might (mistakenly) be called “rational” given certain individual preferences, observations of individual psychology cannot usually provide a proper test of the rational hypothesis. In practice, it has been difficult to verify clear violations of the rational hypothesis. There is often an argument over whether what might be called irrational behavior results instead from rational preferences specific to a particular time and place. For example, although it might seem irrational for a group to demand higher payment for a good already owned than the same group would be willing to pay for the same good not owned, this asymmetry might be explained as rational in view of the consumer surplus in ownership for the group. Because the mechanism traditional economic theory uses to measure preferences is the observation of group behavior, arguments about whether the behavior of a particular group is rational or irrational can quickly devolve into circularity.

This is not to say, of course, that psychology offers no insight to economics. In fact it seems quite likely now that evolutionary psychology and neurology will offer insights into the origins and explanation of observed group behavior. Such insights are not as relevant, however, to the problem of measuring and managing large groups over long periods of time. Systems theory takes a different approach to rehabilitating economic theory.

So the answer is not really. Theories of psychology are very useful and important for business people and leaders in general. But they are not the kind of theories designed to address large-scale, long-term behavior of groups. This is what economics and rational expectations theory has done for the past few decades. It has failed us because behavior that was once uncorrelated, and hence gaussian, has become more interdependent, and hence fat-tailed.

The way forward for us is to recognize that rational expectations are correct only at certain limits, namely, of ergodicity and uniformity in individual preferences. Rational expectations theory is a mean-field approximation. But when we have more granular data about individual dynamics, there's no reason for us to be content with mean-field approximations. Micro and macro theorists need to build nonequilibrium statistical models that are capable of producing both the gaussian and the fat-tailed dynamics that are observable in real markets. Systems theory represents the set of mathematical models that are most narrowly tailored to this goal.

Robust dynamic classes revealed by measuring the response function of a social system

From Riley Crane and Didier Sornette:

We find that most videos’ dynamics (≈90%) either do not experience much activity or can be described statistically as a Poisson process (verified using a Chi-Squared test). This large set of data is consistent with the endo-subcritical classification. For the remaining 10% (≈500, 000 videos), we find nontrivial herding behavior that accurately obeys the three power-law relations described above. Characteristic examples of endogenous and exogenous dynamics are shown in Fig. 3.

Via Computational Legal Studies blog.

By what mechanism does traffic transition form Poisson to Power-law statistics? Click on the "syncronization" category at on the right to see some speculations. It seems that spatially dispersed clusters are syncing up. In the exogenous case, that's in response to a driving force -- new information. In the endogenous case, the mechanism is more subtle, and may be an example of self-organization. See the Zanette and Kuperman summary especially.

Non-convexities in Production

From The Economics of Non-Convex Ecosystems: Introduction by Partha Dasgupta and Karl-Goran Maler:

The price mechanism is especially problematic in economic systems characterised by positive feedback processes. We now know that in such environments it may prove impossible to decentralise an efficient allocation of resources by means exclusively of prices. Efficient mechanisms would typically involve additional social contrivances, such as (Pigouvian) taxes and subsidies, quantity controls, social norms of behaviour, and so forth. This was proved formally in a justly famous article by Starrett (1972), who showed that for certain types of non-convexities associated with environmental pollution, a competitive price equilibrium simply does not exist: markets for pollution would be unable to equate demands to supplies.

The discussion in Section 4 demonstrates a link between gauge theory in physics and non-convexity theory in economics. Dasgupta and Maler "without rigorous justification" associate investment I with the time-derivative of capital stock K. Using the language of gauge theory, one would say that K is a "conserved current." When local symmetries of the system are broken over time or across space, such currents are no-longer conserved.

Bleg for Economists: Empirical Tests of the Rational Hypothesis

Folks, aside from the lab experiments (which I know produced mixed results), what empirical evidence do we have of the validity of the rational hypothesis? As best I can tell, the last work done in this regard were the nonparametric tests of WARP by Varian and Bronars in the 1980s.

Those studies, of course, were of aggregates of household consumption over long time periods. Has anybody tested the rational hypothesis against revealed preferences outside a laboratory by tracking individual consumer behavior?

The question comes to this: Is there a data set for a large population that tracks the time-series of acts of consumption for each individual in the population over some period of time?

If the rational hypothesis is applied to predict how individuals will behave, then it should be based on empirical data about individual behavior, should it not? Where is that data?

Problems with Models of Intertemporal Choice

From The New Palgrave Dictionary of Economics:

For the exponential discount function, the discount rate is independent of the horizon, τ. Specifically, the discount rate is –ln(δ) and the discount factor is δ... The exponential discount function also has the property of dynamic consistency: preferences held at one point in time do not change with the passage of time (unless new information arrives).

...but new information arrives all the time Does it make sense for us to continue to base economics on an assumption that new information arrives only rarely?

The Problem with the Principle of Revealed Preferences and a Suggested Solution

In Chapter 5 of the Foundations of Economic Analysis, Paul Samuelson sets forth what he calls "The Pure Theory of Consumer's Behavior," which he identifies with the analysis of utility. According to Samuelson, many economists "separate economics from sociology upon the basis of rational or irrational behavior, where these terms are defined in the penumbra of utility theory. It would seem extremely important, therefore, to know clearly what is contained in the conventional utility analysis, if only to understand the consequences of denying its validity." (at 90)

What is utility? It is not happiness in Bentham's sense, or welfare in some general sense. Nor is it a physiological or psychological. "Originally great importance was attached to the ability of goods to fill basic biological needs," writes Samuleson, "but in almost every case this view has undergone extreme modification."

Why? Because of "the recognition that a cardinal measure of utility is in any case unnecessary; that only an ordinal preference, involving 'more' or 'less' but not 'how much,' is required for the analysis of consumer's behavior."

This is a weak explanation for why consumer behavior should be considered independent of psychology or biology. Why would biology or psychology demand that preferences be cardinal rather than ordinal when a measure of consumer behavior does not? Samuelson does not say. This makes the fact that evolutionary biology, social psychology, and sociology have made a come back recently less surprising, does it not?

But how does this utility analysis of consumer behavior work then? Starting from the assumption of ordinal preferences, Samuelson postulates the existence of a utility function. To help the reader in visualizing his argument, assume for now that there are only two types of goods or activities, A and B, that a consumer can choose. If we assign the quantity of good or activity A to the x-axis and the quantity of good or activity B to the y-axis, then the utility function would be a set of lines in the x-y plane. None of these lines cross; each represents a set of x,y pairs (or bundles of A and B) which produce the same amount of utility for the consumer. The lines are called "indifference curves."

How does this get us to a theory of consumer behavior? We get there with just two additional assumptions:

The utility analysis rests on the fundamental assumption that the individual confronted with given prices and confined to a given total expenditure selects that combination of goods which is highest on his preference scale.

(at 97) In other words, given a budget, a consumer behaves so as to maximize her utility by selecting the bundle of goods that includes as many of the goods that she ranks highly as possible.

There are two assumptions here. First, that the consumer maximizes utility. Second, that the particular bundle of goods selected reveals that the consumer prefers that bundle over any other.

As one might guess from the title of this post, I see problems with one of these assumptions, and it is not the assumption of maximization. As Richard Feynman has noted, for example, the behavior of many systems (human and non-human) can be described as a constrained extremization. It would be more surprising to find out that consumer behavior does not maximize or minimize some quantity.

The problem is with the principle of revealed preference. What the principle suggests is that the preferences (and hence utility) of a consumer can be measured indirectly (i.e., revealed) by observing what bundle of goods is preferred. The problem is that, by construction, a preference is revealed only when a transaction occurs. Although everybody has experienced an increase and decrease in preferences for particular goods over time (such as hunger and thirst), the principle of revealed preferences gives us no way to measure how preferences evolve in time between transactions. Relatedly and in addition, to make a complete "measurement" of consumer preferences for an entire set of goods, one must assume that the ranking of preferences does not change over time (or at least that it changes in deterministic ways). Otherwise there is no logically coherent way to rank the entire set of preferences.

Why do economists make these assumptions? They assume revealed preferences because there are few obvious alternative methods for measuring preferences. They assume time-independence of preferences because there is no other way to measure the entire preference function. Ultimately, they make both of these assumptions because with them, one can make reasonable forward-looking forecasts about how an economy will change as a result of shifts in supply or demand for a particular good. Reasonable, but not perfect!

There are alternatives for measuring preferences. On this blog a year ago I started writing about one. Consider as an alternative to the ordinal ranking of preferences a cardinal measurement of preferences. The preferences of a consumer for a particular good could be measured by counting how often the consumer consumed the particular good within a specific window of time. This measurement of consumer preference would take units of frequency. So if a consumer ate 5 apples in a day, we would say that the consumer had preference 5 apples per day. If the consumer ate only 1 orange per week, then we would say that the consumer had preference 1/7 oranges per day.

Notice how this alternative method of measuring preferences is consistent with the principle of revealed preferences in the sense that within a particular window of time we can rank preferences for each consumer. Within the window of the one week observed, the consumer preferred apples to oranges. If the preferences of this particular consumer are stable over many weeks, then the preference function observed through revealed preferences will be either almost or exactly the same as the preference function measured directly through frequency of consumption and then ranked. This condition of stability is called "ergodicity" by physicists.

But notice also how this method of measuring preferences provides additional flexibility. In particular, the model permits for us to treat preferences as varying with the time-scale of observation. Over the course of a month, consumers might not have much interest in oranges. But over the course of a year, there may be a more substantial preference for oranges.

In addition, the alternative method of measuring preferences permits us to get more information out of the fact of exchange. Whereas before an exchange was our only source of information about the preferences of the parties to the exchange, now with our independent measurement of preferences we can use the fact of exchange as an additional source of information.

Let's say Charlie sells A to Warren for B. If we know that Charlie uses an A / week and B / day, then we can infer that Charlie probably has either some inventory of B or expects to get access to more B within the next day. The fact of exchange tells us about the model that Charlie and Warren have in mind in forecasting their future needs based on their memory of the history of their preferences. In other words, the fact of exchange in view of measured frequencies of preferences still tells us something about their subjective valuation, but it tells us something more subtle than that Charlie prefers B to A and Warren vice-versa. It tells us how often Charlie and Warren expect to use A and B given their current inventory and history of consumption.

For earlier Broken Symmetry posts on related topics see here and here.

In a nutshell, my suggestion is to replace the principle of revealed preferences with a hypothesis of periodicity. The principle of revealed preferences would still obtain in the limit of ergodicity.

Time-Frequency Distributions and the Characteristic Time-Scale of Economic Equilibria

Over the weekend, it sort of dawned on me that in looking at Google Insights trends (see, e.g., here), what one is really looking at is a time-frequency distribution.

Over some time period that Google does not specify, Google bins the numbers of searches made in geographical regions, normalizes the number of searches by the total volume of searches within the region, and then rescales the results so that the peak value within the time-window displayed is set equal 100, and other values proportional to percentages of that peak.

As a consequence of the normalization and rescaling it is impossible to put units of frequency on the y-axis in Google Insights charts. The information needed to put units of frequency on the y-axis would be destroyed by either step independently, but is long gone after both.

But even without units of frequency, we can make some qualitative analysis of what these Google Insights tell us about the frequency of demand for particular goods.

For example, when one observes seasonal variations in Google Insights charts, that's equivalent to observing an increase in the frequency of demand. I can't tell you exactly how much faster because only Google knows that. But I can tell you that if Google Insights shows a seasonal variation in searching for your product, then you either need to have inventory in storage or your production cycle has got to speed up to match. The shortages or surpluses that result from the mismatches in the frequency of demand and frequency of supply cause prices to fluctuate; market price signals thus perform a frequency and phase locking function.

Time-frequency distributions are used by engineers, chemists, and physicists to study how the frequency content of a time-varying signal changes in time. Within an economy, variations in the time-frequency distribution of demand show that there are different characteristic frequencies to demand for a particular good. As a result, there is no single economic equilibrium, which is valid at every time-scale. Rather, equilibrium price will depend on the the time-scale over which supply and demand are observed. Supply and demand have to be orchestrated within a hiearchy of longer and longer time-scales in order to avoid systematic errors in frequency and phase-locking.

Tyler Cowen on Fischer Black and Correlated Risk

The key question about the current financial crisis is how so many investors could have mispriced risk in the same way and at the same time. This article looks at the work of Fischer Black for insight into this problem. In particular, Black considered why the “law of large numbers” does not always apply to expectations in a market setting. Black’s hypothesis that a financial crisis can arise from extreme bad luck is more plausible than is usually realized. In this view, such factors as the real estate market are of secondary importance for understanding the economic crisis, and the financial side of the crisis may have roots in the real economy as a whole.

Download available here.

Large-scale structure of time evolving citation networks

By E. A. Leicht, Gavin Clarkson, Kerby Shedden, and M. E. J. Newman.

As the plot shows, there is a marked trend for the average age to increase in step with the passage of time. This is precisely the behavior one would expect if the top authorities in the network are remaining the same as time goes by. Every once in a while, however, the plot shows a sudden and precipitous drop in the average age, indicating that a much younger set of vertices have, in a short space of time, taken over as the new leaders in the authority score rankings. Thus the plot indicates a repeated pattern in the evolution of the network in which a certain set of vertices—certain cases considered by the Supreme Court—remain the top authorities for substantial periods of time before being swiftly replaced by a different set. One example of such a turnover can be seen in Fig. 8 around 1900 and a smaller one around 1940, dates that, as we have seen, correspond roughly to the beginning and end of the Lochner era. Another very large dip in the curve occurs around 1970.

Available here. The authors don't say so in their analysis, but the associated graph suggests a mechanism of self-organized criticality.

Time and Turbulence

Although nobody has done it yet for economics, the connection between microscopic and macroscopic dynamics is old news in physics. Over fifty years ago, and with antecedents even earlier, mathematicians and physicists discovered that by coarse-graining certain variables that describe how units of a system interact, the macroscopic dynamics of a large system could be derived from microscopic variables. The technique is called renormalization, and the wikipedia entry on renormalization group is decent enough.

To visualize how this works, it helps to picture the microscopic system as a network of nodes distributed spatially, with one or more vectors associated with each node denoting how some observable quantity flows from one node to another. In general, the magnitude and direction of these vectors may vary, but in the simplest models (such as the Ising model), there is only one vector and it has the same magnitude at each node although it varies in direction.

With nodes distributed along only one-dimension, feedback doesn't usually develop within the network. Interactions, by construction, are limited to nearest neighbors in such a network. (Doesn't this explain why long supply chains can be unstable?) The exception is for when the end of a one-dimensional chain is linked to the beginning, creating a ring. With that geometry, even a one-dimensional network can display non-transient periodicities.

But in many real networks, such as our economy, the linkages between nodes are multi-dimensional. Each link in the supply chain for a particular good is also part of multiple other supply chains for its inputs and outputs. This multi-dimensional network permits all sorts of stable and unstable feedback loops to form.

Renormalization entails combining smaller units of a system with a defined symmetry into larger units with the same symmetry. Iteration of this process eventually results in either a cancelling out or a convergence of microscopic variables. Usually, there is an "order parameter" associated with one of the microscopic variables that will end up in one of three states as the result of these iterations when it is "coarse-grained" to macroscopic scale -- perfect order, perfect randomness, or fractal (i.e., self-similar) structure at a very specific point in between. What is less well-understood is what drives the transitions from order to disorder, and especially the fractal state in between.

The dynamics of flow -- i.e., fluid dynamics -- provides some interesting clues. In fluid dynamics, physicists have used the momentum of vortices to model turbulence. Unless you already know a little physics, this may be hard to appreciate, but for every dynamic loop in space, there is an associated angular momentum, perpendicular to that loop. The sums of the angular momenta of vortices have been used in what is to date perhaps still the most successful model of turbulence in a fluid.

The formation of vortices (and so, by analogy, feedback loops) in fluids leads to all sorts of interesting dynamical patterns. In the past, it's been very difficult to study fluids such as water experimentally, so other fluids such as electron plasmas, have been studied instead. Among other interesting phenomena, one may observe "vortex crystals" form within a fluid as a turbulent flow steadies over time. These may be an example of what Ilya Prigogine calls dissipative structures. Feedback loops seems to be a key prerequisite to their formation.

At this point, we're already in deep waters, but if you've made it this far, then bear with me just a little further. From Kolmogoroff's theory, it appears that the vortex structure of turbulent flow bears similarity to the synchronized or self-organized critical structure observed in the brain during gamma wave resonance. Both the spatial and temporal structure of the vortex vectors associated with turbulent flow and the signal transduction vectors associated with gamma wave resonance obey power laws to a reasonable approximation.

Aside from the fact that such different phenomena can be approximated using similar mathematical models (not so remarkable to many physicists, really), this connection is remarkable for what it suggests about our perception of time.

Gamma waves are associated with a particular state of consciousness. A person can train to enhance the onset of gamma wave resonance. In fact, there are zen monks who have been experimentally verified to have this ability. A fact about zen meditation that might be familiar only to those who have practiced it is that it changes, at least temporarily, one's perception of time. As Csikszentmihalyi describes with respect to flow, the perception of time while in meditation may dilate or contract, but it does not remain steady.

The implication is that our perception of time is influenced by the structure of feedback loops within our brain. In particular, we lose our sense of the flow of time when these feedback loops are organized into a particular, self-similar structure that can be approximated (like turbulence) with power laws.

That's enough speculation for today. But as an aside I can't help wondering whether the measurement of time more generally cannot be said to depend on the underlying structure of feedback loops within the dynamical system studied. Doesn't the flow of time itself depend on the way in which the ticking clocks used to measure it out are linked to one other spatially? Does time flow because of an increase in turbulence?

Gravity causes convective flow. Convective flow can lead to turbulence. Thus, gravity causes time to flow?

Note that Roger Penrose has also speculated on how gravity plays a role in constituting our consciousness. Commenters more versed in the Second Law than I are welcome to rebut these speculations below!

Spontaneously Broken Continuous Symmetries in the Preference Functions of Economic Actors

Suppose we model an economy as a network of individual actors. Each actor is a node in the network. Each has a bundle of goods and a preference function associated with it. Consumption and production activities result in a transformation of the bundle. Exchanges result in a flow of goods through the network.

To a good approximation, at any given moment in time if we looked at the preferences for a particular good at each node, we might observe a continuous symmetry -- i.e., the relative preference for the good at each node might be approximately the same throughout the nework.

Now imagine that we watch what happens over time if we permit for a change in preferences to result from a (for now assume exogenous) new transormation discovered at a particular node. Suppose, for example, that one node in the network discovers a way to make a similar quality good using fewer inputs. At that node, the preferences for the particular good produced are going to diminish relative to other goods because the same (or similar good) can be produced at lower opportunity cost.

And now suppose that other nodes in the network can somehow (via mirror neurons or language) learn about this new means for producing the good so that both the knowledge and the change in preferences can propagate through the network.

Such propagation is arguably an example of the spontaneous breaking of a continuous symmetry in preferences. The observable result would be an oscillation -- a goldstone mode.

In general, not all continuous symmetries can be spontaneously broken to result in a goldstone mode. Under the Mermin-Wagner Theorem, continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions less than or equal to 2. (Isn't this an explanation for why a one-dimensional series of price changes does not have a finite average?) But apparently, so long as the nodes in our network can interact in more than two dimensions (which they can since our preferences, in general, can have as many dimensions as there are goods), then there can be goldstone modes.

Let's take this a step further by imagining that instead of each node being connected only to a neighborhood of other nodes, each node is connected to all other nodes -- i.e., that our network is all-to-all. How might this be accomplished? By designating a central place for exchange -- a market. Now when a new preference is discovered, regardless of whether the new preference is communicated to any other node directly, the new preference will be reflected by a new kind of exchange in the market, which is observable (by construction) by all nodes in the network.

How will propagation of this new preference take place in an all-to-all network? It depends on whether the new transformation that resulted in the new preference can be "reverse-engineeered" by observing the new exchange alone. If so, then the new preference may be adopted instaneously, and a new equilibrium reached within the time needed for the new kinds of exchanges to be completed (although this will depend on the frequency of previous consumption, production, and exchange activities). If not, then the market mechanism will not result in a global approach to the new equilibrium. Rather, alternative mechanisms for communicating the new transformation will be required.

In addition, the time required for frequency and phase matching a new exchange within the framework of existing exchanges among the nodes should not be overlooked. Especially if the new transformation occurs at low-frequency, it might be a long time before the network transitions to the new mode of operating through the necessary exchanges required.

Robert H. Frank's "Nonpositional Goods" are the Gauge Bosons of Local Symmetries in Economics

In "Positional Externalities Cause Large and Preventable Welfare Losses," Robert H. Frank notes how so-called "positional goods" may effect a redistribution of wealth in society by drawing resources into their creation that might be more efficiently allocated to other goods.

Earlier posts here and here considered the relevance of gauge theory to neoclassical economic models and concluded that invariances such as the Coase Theorem and the Miller-Modigliani theorem are conservation laws that are linked by Noether's Theorem to important symmetries in the neoclassical model.

I've been chewing this over for a while. An important question raised here is "What are transactions costs?" Somebody unfamiliar with the Coase Theorem or New Institutional Economics might naively believe the category limited to brokers' or attorneys' fees. They are not. Coase understood these costs to include, for example, the costs of searching for and identifying potential counterparties in addition to such fees. In fact, a major project for research within the field of New Institutional Economics seems to be getting precise in categorizing transactions costs.

On my view, the difficulty with categorizing transactions costs can be traced back to the fact that one person's "costs" are another person's revenue. In other words, transactions costs are deeply interwined with the symmetries or asymmetries in the bundle of goods and preferences held by potential counterparties.

Physicists have developed a very useful tool for categorizing particles based on the symmetries of the fields through which particles interact. This is gauge theory, of which Yang-Mills theory, which gives us the Standard Model, is the most well-established.

But before Yang-Mills theory there was Weyl's classical field theory. Classical field theory in physics actually bears much in common with neoclassical theory in economics. Both theories assume that symmetries in key variables are global. In neoclassical theory, if the value of some currency is inflated, then it may seem to each individual like they are richer, but because the currency value is inflated for everybody, buying power -- the value of the currency -- is the same.

By contrast, Yang-Mills (and in general any non-abelian gauge field) Theory permits for some symmetries to exist only locally. Many interesting consequences flow from the fact that global symmetries may be broken. One is path-dependence: The order in which interactions take place in Yang-Mills Theory matters to the ultimate configuration of the system. In fact, as Yang himself put it, when global symmetries are broken, "local symmetries dictate interactions".

Analogy to Yang-Mills Theory might bear fruit for economists. In particular, the methodology of categorizing interactions through local symmetry breaking could be very useful in formulating a more precise theory of transactions costs.

I will have more to say about this over time. For now, let me observe simply that most goods are produced and exchanged in discrete units. In physics terms, this means that the field of value in these goods is quantized. According to neoclassical economic theorists, this quantization shouldn't matter because goods can be exchanged indefinitely until the closest match to equilibrium is achieved.

That isn't what happens in practice, of course. To wit, what Robert H. Frank calls "nonpositional goods" are the analogs to gauge bosons of non-abelian gauge theory. Because the field of values for a good is quantized and because only local symmetries are conserved for the good, the lowest energy eqilibrium may never be reached.

But identifying "transactions costs" as the gauge bosons that result from local symmetry breaking is probably a useful insight in itself considering how difficult it has been to be precise about transactions costs in the past.

Frequency Matching Supply to Demand: What CEOs Could Learn from Surfers

In any market, demand comes in waves. There are two reasons for this: (1) Each consumer who is part of the demand takes a period of time to consume; and (2) After consuming, each consumer is going to wait a period of time before consuming again, if at all. In mathematical terms, for a given population of consumers, there is a frequency distribution characterizing the average period between acts of consumption. (A frequency is simply the reciprocal of the period of time between acts of consumption.)

Of course, if you're looking at a large enough market, or averaging over a long-enough period of time, these frequency distributions might not be obvious. Rather, the quantity of goods in demand within each period of time may appear steady. Demand often appears steady because for many goods the frequency distribution of consumption is stationary (i.e., the frequency distribution itself doesn't change in time) and the phase (i.e., the amount of time between two acts of consumption by different consumers) is uncorrelated. But this stationarity and asynchronicity is often dependent upon the time-scale or size of the population measured.

For example, if one looks at the worldwide demand for oranges in the last 90 days, demand might appear steady as it does here. But if one looks at the worldwide demand for oranges over the past 5 years, then the periodicity in demand because obvious. See here. Oranges are in season in the winter, and demand for oranges has over time synchronized with orange season. Some of you are wondering about why the southern hemisphere doesn't erase this periodicity in demand. It does, partly. Go look at New Zealand and Australia and see how the demand in these countries spikes in the middle of the year. The southern hemisphere simply has a smaller population!

If you work in marketing, especially at a consumer Internet company, probably none of this is surprising to you. You see waves of customer interest come and go all the time. What do you try to do in response to those waves?

The same thing that surfers do when a swell approaches shore. You try to position yourself close to where the wave is going to peak -- or before it if possible -- and then match your momentum to the wave so that it picks you up, and carries you with it.

In physics terms, by frequency matching (the momentum of a wave is mathematically equivalent to a spatial frequency), you maximize the amount of energy that can be exchanged between the wave and you. If you don't paddle hard enough, in the right direction, at the right time, or in the right spot, then you'll simply be left behind. By contrast, if you can match the momentum closely enough, then you will resonate with the wave -- the wave will push you forward as you push back on the wave. Momentum-matching results in far more fun.

What happens if you paddle too fast? If you paddle way too fast, you'll paddle right over the wave in front of you. For anybody who has actually surfed before, this is hard to imagine because in practice nobody paddles that fast. But the point to see is that if you paddle too fast, then you'll end up oscillating faster than the waves of demand -- overshooting and undershooting with too much or too little inventory when the waves of demand arrive.

Too fast, oscillation. Too slow, left behind. Just right means you catch the wave, and your supply grows at the same rate as the wave of demand. This rate of growth is both efficient and sustainable since, by definition, it matches demand.

The mathematical formalism is equivalent to that used by electrical engineers in describing the waves of voltage and current in a circuit. Momentum-matching the surf, frequency-matching demand, and impedance-matching an input signal are all the same phenomenon as far as the formalism is concerned.

What role do market prices play in matching supply to demand? Market prices are a mechanism for active impedance matching. When the frequency of demand is steady in time, then market prices synchronize the phase of supply and demand cycles. When both the frequency and phase of demand are non-stationary, market price signals synchronize both. I will have more to say about this, but the synchronization of supply and demand at large scales can be a cause of either incredible stability or bubbles and crashes. That's what control theory is for figuring out.

Periodicity or Rationality?

From today's WSJ:

Optimism regarding industrial metals rests on similarly fragile foundations. Bellwether copper is up 43% since late February. This head fake comes courtesy of China cannily stockpiling inventories of raw materials like copper and iron ore even as demand contracts virtually everywhere else.

Stockpiling, however, isn't consumption. The surge in Chinese imports hasn't been matched by increases in industrial production, suggesting metals are going into warehouses, not factories.

The actions of the Chinese could be explained as rational in response to their forecasted needs. Alternatively, we could just note the amount of commodities flowing into China per unit time and how that amount changes in time. Does it matter whether the latter is explained as a rational choice?

How Liquidity Constraints Alone Can Produce Price Bubbles

This post requires readers to have some background in mathematics, and complex analysis in particular. The point is to give a very succinct and precise explanation for how and why price bubbles can result without any assumption of rationality or irrationality. Bubbles can result from liquidity constraints alone.

For purposes of the argument, I make the following definitions of key variables:

Let q be defined as the discrete time-varying quantity that represents the volume of transactions that can occur within a market within a particular window in time. If it helps, think of q as a measure of aggregate current inventories. Let's set units of volume as dollars (although any fungible good might work). Then the units of q are dollars per unit time ($/day, for example). The derivative dq/dt thus specifies how the dollar volume per unit time is changing.

Let c be defined as the bid-ask spread in volume -- i.e., as the difference in dollars between the volume of goods offered for sale and the quantity of goods asked for purchase at a given moment in time. This bid-ask spread can be read off the markets in real-time. For purposes of this argument, we will assume that it does not vary much in time relative to q -- i.e., that dc/dt << dq/dt.

Here comes the key assumption for my argument:

Assume market price p is proportional at each moment in time to the volume available for transactions and inversely proportional to the bid-ask spread, i.e., that:

p = q / c

If we assume also that the total number of transactions that occur within the market within the time window within which q is measured is a conserved quantity (i.e., that inventory q is not created or destroyed within the relevant time frame), then we can define the "current" of transactions as:

i = -dq/dt

Now let L be defined as the amount of time it takes to complete a transaction at a given price level within a unit of time, so that:

p = -L di/dt

Finally, assume that some transactions costs R exist, but do not vary much on the relevant time-scale. Then p = i R.

Electrical engineers and physicists will recognize this as the model used for an LC circuit. I've even chosen the notation to be the same to avoid confusion in applying the model

These quantities together constitute a partial differential equation (PDE) that can be solved for p. The easiest way to do that is to treat p, q, and i as complex variables with a magnitude and time-varying phase. Then after a Laplace transform of the PDE, the complex transfer function is found by algebra:

H(s) = R / [R + (sL - 1/sC)]

H(s) peaks when its denominator gets closest to zero, i.e., when:

sL - 1/sC = 0

Or when s = 1 / sqt(LC)

What this means is that for a given volume of inventory (take, for example, the money supply M1), there is a characteristic frequency for fluctuations in price, which is determined by the liquidity variables L and c. Let's say that somehow the price could be changed exogenously (for example, by increasing or decreasing M1 with interest rates). When the period of time required for a cycle of increase and decrease matches 1/sqt(LC), then price will peak. So long as L and c can accommodate the dramatic fluctuations in p, everything will be fine. But sometimes the system will blow up. For example, the market may not be able to accommodate the total inventory available for sale (i.e., c may break down) or the price of the transaction may require additional transactions costs in time or money (R or L may go nonlinear).

Something like this just happened in the U.S.

Douglass North on What We Don't Understand

"Economics, political science, sociology ... are static disciplines... [that] assume[] away all of the interesting things we want to understand... What's missing? Belief. Time. Culture."

Broken Symmetry readers should take note that "non-ergodicity" is a characteristic of dynamical systems with spontaneously broken symmetries.

You can download a powerpoint presentation in which North makes similar points here.

And here is an academic paper on the same subject.

For more on these topics, see an earlier post on The Three Laws of Performance here.

Would it be fair to call an "open access" state a manifestation of Csikszentmihalyi's flow at the level of organizations and organizations of organizations?

Harald Weinrich on the Connection Between Time and Money

From On Borrowed Time:

In the credit system as it functions in the modern industrial world, money is closely connected with time, and more precisely with the deadline-period. That for society this is a daring, risky, fortune-hunting connection is already shown by the fact that Benjamin Franklin's well-known saying, "Time is money," is not reversible. Although time (in the form of life-time) can easily be exchanged for money, according to the rules of labor and commerce, the money received cannot be re-exchanged for time.

Seneca on the Price of Life

From Seneca On the Shortness of Life:

That advocate is lionized throughout the whole forum, and fills all the place with a great crowd that stretches farther than he can be heard, yet he says: "When will vacation time come?" Everyone hurries his life on and suffers from a yearning for the future and a weariness of the present. But he who bestows all of his time on his own needs, who plans out every day as if it were his last, neither longs for nor fears the morrow. For what new pleasure is there that any hour can now bring? They are all known, all have been enjoyed to the full. Mistress Fortune may deal out the rest as she likes; his life has already found safety. Something may be added to it, but nothing taken from it, and he will take any addition as the man who is satisfied and filled takes the food which he does not desire and yet can hold. And so there is no reason for you to think that any man has lived long because he has grey hairs or wrinkles; he has not lived long—he has existed long. For what if you should think that that man had had a long voyage who had been caught by a fierce storm as soon as he left harbour, and, swept hither and thither by a succession of winds that raged from different quarters, had been driven in a circle around the same course? Not much voyaging did he have, but much tossing about.

. . .

Life is divided into three periods—that which has been, that which is, that which will be. Of these the present time is short, the future is doubtful, the past is certain. For the last is the one over which Fortune has lost control, is the one which cannot be brought back under any man's power. But men who are engrossed lose this; for they have no time to look back upon the past, and even if they should have, it is not pleasant to recall something they must view with regret. They are, therefore, unwilling to direct their thoughts backward to ill-spent hours, and those whose vices become obvious if they review the past, even the vices which were disguised under some allurement of momentary pleasure, do not have the courage to revert to those hours. No one willingly turns his thought back to the past, unless all his acts have been submitted to the censorship of his conscience, which is never deceived; he who has ambitiously coveted, proudly scorned, recklessly conquered, treacherously betrayed, greedily seized, or lavishly squandered, must needs fear his own memory. And yet this is the part of our time that is sacred and set apart, put beyond the reach of all human mishaps, and removed from the dominion of Fortune, the part which is disquieted by no want, by no fear, by no attacks of disease; this can neither be troubled nor be snatched away—it is an everlasting and unanxious possession. The present offers only one day at a time, and each by minutes; but all the days of past time will appear when you bid them, they will suffer you to behold them and keep them at your will—a thing which those who are engrossed have no time to do. The mind that is untroubled and tranquil has the power to roam into all the parts of its life; but the minds of the engrossed, just as if weighted by a yoke, cannot turn and look behind. And so their life vanishes into an abyss; and as it does no good, no matter how much water you pour into a vessel, if there is no bottom21 to receive and hold it, so with time—it makes no difference how much is given; if there is nothing for it to settle upon, it passes out through the chinks and holes of the mind. Present time is very brief, so brief, indeed, that to some there seems to be none; for it is always in motion, it ever flows and hurries on; it ceases to be before it has come, and can no more brook delay than the firmament or the stars, whose ever unresting movement never lets them abide in the same track. The engrossed, therefore, are concerned with present time alone, and it is so brief that it cannot be grasped, and even this is filched away from them, distracted as they are among many things.

. . .

And so when you see a man often wearing the robe of office, when you see one whose name is famous in the Forum, do not envy him; those things are bought at the price of life. They will waste all their years, in order that they may have one year reckoned by their name. Life has left some in the midst of their first struggles, before they could climb up to the height of their ambition; some, when they have crawled up through a thousand indignities to the crowning dignity, have been possessed by the unhappy thought that they have but toiled for an inscription on a tomb; some who have come to extreme old age, while they adjusted it to new hopes as if it were youth, have had it fail from sheer weakness in the midst of their great and shameless endeavours. Shameful is he whose breath leaves him in the midst of a trial when, advanced in years and still courting the applause of an ignorant circle, he is pleading for some litigant who is the veriest stranger; disgraceful is he who, exhausted more quickly by his mode of living than by his labour, collapses in the very midst of his duties; disgraceful is he who dies in the act of receiving payments on account, and draws a smile from his long delayed heir.

Comments on Julian Barbour's Essay The Nature of Time

Barbour won an essay contest on the subject of time. See here.

Barbour's clever argument is that our experience of time is always mediated through a comparison of the relative movement of two systems. A clock is simply a dynamical system that is phase-locked with another dynamical system, both of which satisfy a principle of least action.

In a key passage, Barbour takes issue with Einstein's definition of a clock, which is (basically) anything periodic that passes through an identical phase. Here are Barbour's criticisms:

First, no system ever runs through identical phases, so this is an idealization that hides the true nature of time; it lacks Poincare's insistence, which I have repeated, that only the universe and all that happens in it can tell perfect time. Second, Einstein's clock cannot measure time continuously. It can only indicate that a given interval has elapsed when identical phases recur. It can say nothing about the passage of time in intervals within phases.

Barbour's first point is true, but irrelevant to distinguishing between Einstein's definition of time and his own. Of course Poincare is right that the entire universe would provide the best frame of reference for time. But we cannot measure the entire universe, only a part. Is it possible, then, to know whether a given system obeys the principle of least action with perfect accuracy? Given these practical limitations, there is no obvious benefit that I can see to denying any reasonably periodic recurrence of phase the label of "clock." It is a matter of common knowledge that the things are not perfectly accurate.

Barbour's second point is partly true and partly false. It is true that Einstein's clocks could not measure time continuously. But this is just another way of making his first point. Again, until we become gods, we must content ourselves with approximations of whatever we seek to measure.

But it is plainly false that Einstein's clocks can say nothing about the passage of time in intervals within phases. They can say something about anything that happens within their period of oscillation at an interval of twice their period or longer.

The philosophical point here is that we ought to forget about deciding whether or not time exists in some metaphysical sense, and get down to the business of making use of it in a physical one.

Or to put it another way, here is Harald Weinrich in On Borrowed Time:

Considered in relation to the body, the sense of time can be either added to the canon of the five senses as a "sixth sense" with the same status as the others, or identified as superior "common sense" (sensus communis, bon sens) that is the psychic coordinator of all the other senses. In either case, the sense of time is a "real," that is, corporeal, sense, because it has its own bodily organ -- according to the older view, the temples, and according to the modern view, the circulation of the blood and its rhythmical pulse beat. Although we generally remain unaware of it, like the other senses, in crises it will be perceived beating on the surface of the body. However, since evenly flowing circulation of the blood is its condition, the sense of time takes on a special anthropological status insofar as it is at the same time an internal sense, as the only one that determines the beginning and the end of human life-time. To that extent we can say of the sense of time, and of it alone, that it creates or abolishes the presuppositions not only for itself, but also for all the activities of the other senses.

To put it still another way, since it is impossible for the future to be determined from the past, we should remain agnostic about applications of our theories that purport to say things about the universe or even about next week.

Gauge Theory in Economics

Here are few additional thoughts on the gauge invariance approach to neoclassical theory.

The Coase Theorem in neoclassical theory plays the role of a Noether's Theorem in classical mechanics. Miller-Modigliani is another Noether's Theorem that could be derived from a gauge invariance in neoclassical theory.

So here's the next question: Are there analogs to non-abelian gauge fields in neoclassical economic theory?

For that to be so, it seems that the order in which exchanges occur within an economy would have to matter to the equilibrium established. Already we are out of neoclassical theory then. But the path-dependence of economic growth is an accepted fact among some economists, so perhaps there are non-abelian gauge fields.

One mechanism whereby path-dependence might be introduced is through changes in cross-elasticity. An exchange of one good can effect the price of another, and cross-elasticity is the measure of such a change. There may be a non-abelian gauge field that dictates the cross-elasticities of demand.

Personally, I'd rather see a model of coupled parametric oscillations applied to understanding supply and demand than an agent-based model. But I don't have the time to really dig into this. And I'm not sure that adding formalism is important here. Douglass North and the other New Institutional Economics scholars seem to be pursuing a nonequilibrium theory of economics without one.