Chaotic Economies and Adam Smith’s Invisible Hand

Chaos, chaos everywhere!      Image Credit: Flickr.com/SantaRosa OLD SKOOL

It is sometimes said that “if the only tool you have is a hammer, then everything you see looks like a nail.” Well, I have a shiny new hammer called chaos theory, and while it is not the only tool in my toolkit for thinking about problems, most interesting problems involving large systems sure are beginning to look like nails to me! Luckily, most large, complicated systems do indeed seem to have aspects of chaotic behavior, and a modern economy is such a system. So let us proceed to bang away on this nail!

The Problem and a Description of a System

In my last two posts I discussed an economy’s mechanisms for balancing the demands by consumers for economic goods and services with their supply by producers. This is the most basic problem that any economy must solve. More specifically, for every good produced by the economy, the economy must make the quantity and price of the good desired by all consumers to be at least approximately equal to the quantity and price of the good that producers are willing to supply, Any imbalance in supply and demand will lead to either shortages or surpluses, either one of which will cause businesses to layoff workers and idle production capacity.

For both neoclassical and Keynesian economists, this mechanism is usually called “Adam Smith’s invisible hand” since it was first described by Adam Smith in his 1776 classic tome An Inquiry into the Nature and Causes of the Wealth of Nations. In describing how producers meet the needs of an economy with their industry, he writes,

… by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain; and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it was no part of it. By pursuing his own interest, he frequently promotes that of the society more effectually than when he really intends to promote it.

The emphasis in this quote is mine. In my last post Economic System States, Feedback Loops, and Adam Smith’s Invisible Hand, I discussed how the invisible hand is a metaphor for the intertwined action in a free-market of the laws of supply and demand and of marginal utility. The qualifier that the two laws must interact in a free-market is absolutely crucial. Only in a free-market would the price for each good be driven to the market equilibrium price where the supply and demand curves intersect. For details on this process, read my last post.

So far, I have described the workings of Adam Smith’s invisible hand from two different perspectives. The first in Adam Smith’s “Invisible Hand” and Evolution looked at an analogy of the invisible hand’s actions with the actions of biological evolution on living species. I concluded the analogy was apt because the invisible hand had aspects of both mutation (the provision of new or improved goods) and natural selection of the fittest (the finding of an intersection of supply and demand curves at a non-zero quantity and a non-zero price). If such an intersection is not found, the product is not selected for survival.

The second perspective was a view of an economy as a homeostatic system in society, with the invisible hand regulating economic behavior with the necessary feedback for homeostasis provided by the law of marginal utility. This picture was examined in the posts Feedback Loops and Economic System States and Economic System States, Feedback Loops, and Adam Smith’s Invisible Hand.

In this essay I will look at an economy as a chaotic system. The chaos in the economy has interesting implications that show why non-free-market solutions to the Economic Problem will never work well. But first we need to be able to describe the state of an arbitrary system at any time. I touched on such a description in How to Solve Problems in Chaotic Social Systems.

In order to describe the state of any system, a person needs to know the values of all variables within the system. In physics one has the example of a many-body problem made up of a large number N of particles, which may be of different species. To describe the physical state of the system, one must know six variables for each particle: the three components of the particle’s position, and the three components of its velocity. What must we know to specify the state of an economy?

Just as in the physical N-body problem, an economy is made up of a large number of interacting “atoms” that are not all the same. These economic atoms are usually referred to as economic agents, and they may be individuals, households, or groups of individuals such as companies and non-profit groups. The current state of the economy can be described by specifying for each agent and for each good provided by the economy the following variables: the quantity and price at which the agent would be willing to supply the good, and the quantity and price at which the agent would be willing to purchase the good. If the number of goods available in the economy is G, we need 4G numbers to describe the economic state of each agent. If there are N agents in the economy, we then need to know a total of 4GN numbers to completely specify the state of the economy. Alternatively, we could say we need to know the position of N points in a 4G dimensional phase space, one point for each economic agent. The reason why each economic agent would have coordinates for both supply and demand is some agents would produce some goods and consume others.

To create a complete chaotic theory of the economy, we would also have to derive the dynamics of the system, which depends on the “forces” of interaction between the economic agents,i.e. how the position of one agent in the 4G dimensional phase space changes the position of another agent at a different position. For now, this is entirely too ambitious a program, especially within the confines of a blog addressed to the general public! Suffice it to say we would have to derive something like Liouville’s theorem in statistical mechanics in physics. Rather than thinking in terms of a collection of economic agents, we would derive a kind of fluid theory of agents, where the presence of agents is described by a density of suppliers and a density of consumers. A change of densities in a region would be the equivalent of shifting and distorting supply and demand curves for the goods in that region. Although the development of a dynamical theory of the economy is far too ambitious at present, we can speculate on some qualitative characteristics of these interactions and their implications. We shall discuss some of these characteristics shortly.

What is Chaos Theory and What Makes Systems Chaotic?

The mathematical theory of chaos is the study of systems that are highly sensitive to initial conditions. Another way of saying this is that chaos theory is the study of complicated systems exhibiting a characteristic called the “butterfly effect”.  This term was coined by Edward Lorenz, a Massachusetts Institute of Technology meteorologist who studied computer simulations of weather. He was renowned for showing how unreproducible these simulations could be because of the chaotic nature of the system. In a 1963 paper “Deterministic Nonperiodic Flow” in the Journal of the Atmospheric Sciences, he wrote:

Two states differing by imperceptible amounts may eventually evolve into two considerably different states … If, then, there is any error whatever in observing the present state—and in any real system such errors seem inevitable—an acceptable prediction of an instantaneous state in the distant future may well be impossible….In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.

Small round-off errors in the inputs of initial conditions would be sufficient to produce huge differences in the system state at a later time. The coining of the term “butterfly effect” came from the title of one of his papers in 1972, “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas?”.

One usual characteristic of chaotic systems is they have a very large number of degrees of freedom, i.e. unconstrained independent variables. In our economic phase space the number of degrees of freedom would be less than or equal to 4GN. Actually, the economic degrees of freedom should be considerably less; we would have to subtract one from 4GN for every constraint on the system. Nevertheless, 4GN is such a huge number for any realistic economy the remainder after subtracting the number of constraints would still be very large. A large number of degrees of freedom increases the number of ways a disturbance in phase space could propagate through the system. Many infinitesimal differences in initial conditions then have the possibility of causing disturbances traveling in many different directions.

In the example of weather systems, the interactions between the molecules making up the atmosphere are almost entirely local, i.e. in order to interact strongly, two molecules would have to have positions very close to each other in physical phase space. The result of one such collision between particles can then be randomized by later collisions with other molecules. Coherent disturbances traveling through the system can be torn into many eddies by the frictional effects of such collisions.

Economic agents should also interact more strongly the closer they are to each other on at least two axes of phase space, namely the quantity and price axes for a particular good g. When the quantity and price for which the supplying agent is willing to sell a particular good g is equal to the quantity and price at which the consumer agent is will to buy g, i.e. where the supply and demand curves intersect, the interaction should be strongest. The actual strength of the interaction in a dynamical theory could perhaps be expressed as some probability for a trade as a function of the difference between the coordinates of the supplier and consumer according to some model. This would change the density of suppliers and consumers in the local volume. That however is not our present concern. What is immediately of interest to us in this essay is that a strong probability of a trade occurs when the coordinates of the supplier and consumer along the axes for good g in phase space are very close; interactions between supplier and consumer are local in the phase space.

The importance of local interactions in a free-market can not be over-estimated. Let us suppose that the interactions were strong and global. In that case the interactions would have the effect of ordering the system globally. There would be no chaotic behavior. With local interactions however, what happens in that part of phase space is almost entirely determined by the density of suppliers and consumers in that local volume of the economy’s phase space, with the effects of other regions in phase space being greatly attenuated. Effects from one region of phase space on a remote region of the space would be propagated by some kind of diffusive process by changing the density of suppliers and consumers in other regions of the space. More concretely, the trade in one kind of good affects trades in other goods, so that a perturbation in the trade and production of one good ripples through the entire economy.

What Does Chaos Theory Say About Adam Smith’s “Invisible Hand”?

What can we expect will happen to an economy if the government strongly perturbs it in some fashion to stimulate the economy? In this essay I will exclude the effects of Keynesian monetary policy for simplicity. If the purpose is to stimulate the economy by increasing aggregate demand for goods and services as widely as possible (that is, a Keynesian stimulus), the government would inject money into projects that employed as large a number of workers as possible. Some of the goods supplied by the economy is the labor of employees for various industries, and the Keynesian stimulus would increase the density of both suppliers (the employees) and consumers (the companies that employ them) for that labor. What the Keynesians most desire is that demand by both companies stimulated by the government and the companies’ employees will translate into an increased density of consumers in a large region of phase space. However, because of the complexity of an economy’s phase space and its population of suppliers and consumers for various goods, the government would find it difficult if not impossible to know if there were a sufficient density of suppliers to satisfy all the new demands. This would almost guarantee a mismatch between the density of suppliers and consumers for these goods, causing all of the problems generated anytime there is an imbalance between supply and demand. The problem the government has is that an initial global stimulation of the economy, especially if it bangs the system hard, can cause extremely unpredictable results at a later time.

What is different if a free-market guided by Adam Smith’s invisible hand allocates resources? In a totally free-market all interactions between suppliers and consumers are local. Any changes in the density of consumers locally will induce local changes in the density of suppliers for the goods involved with greatly attenuated and delayed consequences for regions of phase space far removed. Local changes in suppliers will likewise produce local changes in the density of consumers with similarly attenuated effects far away.

Thinking about economic systems in terms of chaos theory only increases my faith in free-markets.

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