Broken Symmetry

R.I.P. Paul Samuelson, the Isaac Newton of Economics

From Ed Glaeser:

Mathematics doesn’t substitute for genuine insight or knowledge of the real world, but it is necessary if economists are going to craft logically coherent models of complex phenomena. While many mathematical models failed to predict the current downturn, plenty of non-mathematical seers also failed. If we ever do come to understand the vicissitudes of asset markets, that understanding will come in the form of formal mathematical analysis, ultimately inspired by Samuelson.

Glaeser's eulogy also hints at how the time may be ripe for a revision to Samuelson's Foundations:

If people can rank outcomes in terms of their desirability, and if they choose one outcome over another, then people have preferences. (Who can disagree with that?) And if, as Samuelson taught us, those preferences follow a few simple rules, then human behavior can be described with a utility function. That conclusion allows economists to turn to Newton and optimization theory.

The principle of revealed preferences results in a mean-field approximation of individual preferences. The incremental step beyond mean-field theory is gauge theory, and this is why the Coase Theorem and New Institutional Economics are more relevant now than neoclassical theory. What NIE has lacked thus far is a theoretical foundation equal in rigor to the foundations Samuelson laid for neoclassical theory.

Samuelson was like Newton also in that he brought his gifts to bear on solving practical problems for his country. Newton completely redesigned the Royal Mint, caught some of the worst counterfeiters, and advised on how to end a 17th century version of the carry trade, which was draining England's royal coffers. Samuelson led the Federal Reserve through several financial crises and advised countless students in high places on economic issues that affected billions of lives.

Not everybody makes this big a dent in history.

George Soros Explains His Worldview

And it is surprisingly fascinating:

While I was reading Popper I was also studying economic theory and I was struck by the contradiction between Popper’s emphasis on imperfect understanding and the theory of perfect competition in economics which postulated perfect knowledge. This led me to start questioning the assumptions of economic theory. These were the two major theoretical inspirations of my philosophy. It is also deeply rooted in my personal history.

The FT has a whole website up with transcripts from each of his lectures, which are ongoing this week. It's little things like this that make my day.

Transcripts of first three lectures are here, here, and here.

For an earlier post on epistemlogical and ontological questions that offers one version of reflexivity, see here.

In reponse to Soros's proposal that we dichotomize the natural and social sciences, I have this to say: His misinterpretation of Heisenberg's uncertainty principle kills his proposal. In fact it is true of both natural and social scientific theories that our theories influence how reality evolves. It is true too that bad natural science may not have as catastrophic or immediate consequences for us as bad social science. But what we perceive depends in part on our theory of what exists, and those theories are not social. Soros's dichotomy is in fact a continuum.

To wit, from whence did Soros derive his understanding of positive and negative feedback loops? By analogy to the natural sciences. Our exploration of nature is the ultimate origin of all of our theories, bot natural and social.

Mandelbrot and the Glassy State of the Economy

From Fractals and Scaling in Finance, page 367:

Everyone agrees that the distribution of firm sizes is far from Gaussian, hence the proper metaphor is not a "soft," but a "hard" object. This leaves open the question of whether it is a real solid or a glass. Physics describes glasses as being extremely viscous liquids that mimic solids and happen to be particularly difficult to study. In the same way, the lognormal distribution averages mildly in the very long run, but in the short run it mimics the wild randomness of the scaling in a very treacherous fashion. One can call it a sheep in wolf's clothing.

Why is Mandelbrot skeptical of lognnormal distributions of firm size? Because market concentration appears to be a scale invariant quantity -- i.e., the amount of concentration appears not to vary as much with the number of firms in the market as one would expect from distributions with finite variance.

Distinguishing Exogenous from Endogenous Market Crashes

Simoleon Sense links to what could be a useful bit of anlaysis from Sornette et al.:

[W]e have provided an objective test of the hypothesis that the largest negative markets moves are special and form two distinct populations. The exogenous crashes can be attributed to extraordinary important external perturbations and/or news impacting the market. The endogenous crashes can be seen as the natural deaths of self-organized self-reinforcing speculative bubbles giving rise to specific precursory signatures in the form of log-periodic power laws accelerating super-exponentially.

Sornette applied a similar approach to analyzing YouTube video traffic here.

I'm slowly warming up to Sornette's idea of log-periodic scaling. The mathematical part of this model, at bottom, arises from a treatment of fractional dimensions as complex rather than real-valued quantities.

Excellent Sentences from Rick Bookstaber

at Rick.Boostaber.com:

Commodities are not assets. They are factors of production. They do not generate returns, they have no claim on production. They have supply that flows out at a nearly fixed rate short term, and they comprise very small markets compared to the financial markets.

Via Prasanna

Renormalization Group in Macroeconomics

Should I be relieved that this was published in 1996?

A friend clued me in to the fact that macroeconomic theorists were applying the Ising Model a long time ago.

But how far have Prof. Aoki and his students and colleagues gotten with this?

Rizzo on the Austrian Middle Ground

From the ThinkMarkets blog:

The traditional Keynesian view is that if equilibrium obtains it will be a very bad one: depression. But ordinarily the uncertainty of the real world makes the economy prone to all sorts of bubbles, liquidity traps, aggregate demand failures. In other words, it is as if the economy is always on the precipice of some great disaster. Keynes advocated the permanent “socialization of investment” as insurance against this tendency toward disequilibrium leading to horrible equilibria.

The Freshwater view, as I see it, is that we are (almost) always in a state of equilibrium. The only thing that can knock us out is inherently unpredictable shocks, usually of a supply or technological variety. Expectations are usually well-behaved and presumptively “rational” at least in the sense that we do not have to worry about big disturbances arising from this subjective (or “demand”) side of the picture.

The Austrian perspective is more balanced. Austrians have been saying that the current recession was precipitated by excessively low interest rate policy followed by the Fed. They say that the structure of production was distorted in at least two ways. First, investment relatively far from the consumption stage was stimulated as was consumption spending – both at the expense of the middle (a different middle here) and of the replacement/maintenance of capital. (Traditional Austrian view.) Second, low interest rates stimulated greater risk-taking on the part of economic agents. (“New Austrian” view a la Tyler Cowen.) This was not an exogenous change in risk preferences but endogenous, that is, caused by low interest rates, not irrational spirits.

The Ising Model of how Savings and Consumption are affected by money supply and interest rates is an Austrian model in the sense that Rizzo describes here. System-wide shifts from consumption to savings occur as phase transitions because of the (even weak) couplings of decisions by individuals.

The tricky part of any nonequilibrium model of dynamics is modeling how microdynamics aggregate into macrofluctuations when the coupling between individuals (or firms) is stronger. In the Ising model, both interest rates and money supply determine the relative strength of coupling -- low interest rates or high money supply tend to overwhelm the effects of coupling. But the effects of coupling can be modeled in as perturbations to equilibrium.

Austrian models are the middle ground because they provide a theory that connects Freshwater equilibrium models with Saltwater disequilibrium models.

More on the Ising Model of Macroeconomics

A comment on Scott Sumner's *Money Illusion* blog has incited me to make a brief followup.

What the Ising Model of consumption and savings tells us is this:

*If* our neighbors decisions' to consume or save have even a very weak influence on our own decision to consume or save, then the decisions of the entire population will spontaneously correlate when interest rates are near zero and money supply is low.

The model is completely agnostic as to other dependent variables, such as expectations and so on. That's both its strength and its weakness.

A Systems Theory of "Animal Spirits": an Application of the 1D Ising Model

According to Keynes, it is "animal spirits" that drive shifts in consumption and savings:

Even apart from the instability due to speculation, there is the instability due to the characteristic of human nature that a large proportion of our positive activities depend on spontaneous optimism rather than mathematical expectations, whether moral or hedonistic or economic. Most, probably, of our decisions to do something positive, the full consequences of which will be drawn out over many days to come, can only be taken as the result of animal spirits - a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities.

Condensed matter physicists, including Leo Kadanoff, developed a mathematical model that we might find useful in modeling what Keynes described as a "spontaneous urge to action." Although a toy model even for physicists, the 1D Ising model (described by Kadanoff here and by wikipedians here) provides a useful way of understanding the relationship between symmetries in microdynamics and the coarse-grained, macrodynamical variables that are used to categorize the state of a system.

In condensed matter physics, the system is a large number N (at least thousands) of magnetic dipoles arranged on a line, which are exposed both to a heat bath (that guarantees a certain average temperature for the overall system) and a magnetic field (that can be turned on and off to introduce more energy and order into the system). Each of these dipoles can point either up or down (i.e., parallel or antiparallel to the magnetic field).

For economists, the system is a large number N of individuals (or firms), each of which can exhibit either a preference for consumption or a preference for savings. We shall keep track of these preferences by assigning a preference for consumption to a dipole pointing down (i.e., by analogy, against the magnetic field) and a preference for savings with a dipole pointing up (i.e., by analogy, aligned with the magnetic field). Rather than a magnetic field, these individuals are subject to the influence of an interest rate announced by some regulatory authority. Rather than a heat bath, we're going to have a regulated money supply, which guarantees that there is a certain total amount of cash in the system.

Most importantly, the Ising model postulates a link or coupling to the two nearest neighbors. In the condensed matter case, this means that the overall energy of a local region of the system is lower (or higher depending on the type of magnetism) when the individuals in that local region are either all parallel with the magnetic field or all anti-parallel. In our case, this means that the average cashflow in a local region is lower (never higher as it might be in the condensed matter case) when the individuals in that local region are all either saving or consuming. What's the explanation for this coupling? "Animal spirits."

Before proceeding to the most surprising results, let's get a feeling for the microeconomics of our system to make sure they make sense. Given a fixed amount of cash in the system, if we increase the interest rate, we expect the number of individuals expressing a preference for consumption to decrease -- i.e., for their preferences to flip from consumption to savings. By contrast, given a fixed interest rate, we expect the absolute amount of consumption to increase as the amount of cash in the system is increased because the real cost of interest payments decreases with inflation in the money supply.

Now for the results. Without bothering to reproduce the detailed mathematical analysis (the most elegant solution involves use of renormalization group methods), here is what one finds from the exact solution of the 1D Ising model: (the graph is reproduced from the paper by Kadanoff):

The y-axis on the graph corresponds to the average rate of savings (positive values) or consumption (negative values). The x-axis corresponds to the reciprocal of money supply so that the origin of the x-axis corresponds to an infinite money supply. The x-axis is also scaled by a factor, Tc, which corresponds to a critical threshold for money supply to the system (further explained below). The different plots correspond to different values of the interest rate (note that these can go negative, a scenario that has obtained only in Sweden so far).

The amount of insight this diagram gives us into the system is extraordinary. Consider the interpretation of these various plots:

• High positive interest rate: (H=0.5) As money supply decreases from infinite to fractions of its critical value, the average amount of savings smoothly increases from about 75% of the population saving to the point at the top right where everybody in the system is saving.
• High negative interest rate: (H=-0.5) As money supply decreases from infinite to fractions of its critical value, the average amount of consumption smoothly increases from about 75% of the population consuming to the point at the bottom right where everybody in the system is consuming.
• Zero interest rate: (H=0) The average amount of consumption and savings is perfectly balanced, even with an infinite money supply. In other words, when the system is flooded with money and interest rates are held at zero, exactly half of the population consumes and exactly half of the population saves. When interest rates are zero and money is plentiful, individuals basically act as if independent of their nearest neighbors. But note well what happens when interest rates are zero and the amount of cash in the system is withdrawn. At a critical threshold (Tc in the diagram), individuals spontaneously start to align with their nearest neighbors by either saving or consuming -- either is possible -- even when interest rates are zero. This is a textbook example of spontaneous symmetry breaking, which is the phenomenon from which this blog takes its name. The broken symmetry is a consequence of the coupling among nearest neighbors -- i.e., "animal spirits."

I realize that most economists have never heard of the 1D Ising model. But don't you think this kind of model might be useful in understanding macroeconomics? In subsequent posts, I will try to discuss how the Ising model has been extended into higher dimensions -- an important generalization for economics. For the time being, it must suffice to say that this toy model has nonetheless been highly influential among physicists in understanding nonequilbrium dynamics.

Update: A reader comments:

One problem with this model is that as the money supply goes to infinity, you get hyperinflation. This makes interest rates rise sharply. But your model has rates go to zero.

The answer to this comment is that interest rates might still depend on money supply. Referring back to the diagram, we can see that interest rates and money supply are treated as independent variables -- i.e., both are independent of the dependent variable (average consumption/savings). So in the zero interest rate case (H=0), interest rates are held as money supply is increased and decreased to see what happens to savings and consumption. The diagram doesn't make obvious how interest rates might depend on money supply (or vice versa).

Nonethelesss, we can deduce a few things about such a relationship from the diagram. For example, from the diagram it is clear that if we were to set a very high or very low interest rate (i.e., an H near +1 or -1), then everybody would be either saving (H=+1) or consuming (H=-1) regardless of whether the money supply was infinite or zero. In that sense at least, the comment about infinite money supply and hyperinflation being correlated is still consistent with the model. The diagram simply doesn't show how interest rates might depend upon money supply (or vice versa).

But we can develop a more complete answer by peaking back at the condensed matter system that inspired the 1D Ising Model. How does the strength of the magnetic field affect temperature? It is important, in this context, to remember that "temperature" is actually a number assigned to a distribution of energy in the system that has a particular shape. Whether an increase or decrease in the strength of a magnetic field will affect the shape of the distribution depends on the details of the microscopic dynamics. With too much or too little energy available to the system, a change in the magnetic field isn't going to have a big influence on the microdynamics, and hence the temperature will be insensitive to the magnetic field at these extremes.

Translating back into economic terms, this means that when individual decisions about savings and consumption are either perfectly independent or perfectly correlated (conditions which correspond to very high and very low total energy in the system), then a change in interest rates isn't going to have a big influence on money supply.

And does that not seem to be the case in fact?