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?

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