Nonlinear Economic Dynamics

Processes of economic development may be defined as being linear or nonlinear. Linear processes of development are characterized by balancing forces where the economy develops in an incremental fashion. Nonlinear processes of change are driven by destabilizing or compounding positive feedback loops that result in rapid processes of change as the economy moves into a new regime. These rapid and often abrupt processes of change are studied in nonlinear economic dynamics.1
From chaos theory to network theory, complexity science has taught us that these complex systems that we once thought were random are in fact certainly not. We just did not have the tools to model them. We had linear models to describe simple interactions. We thought that if things were not governed by simple symmetries, they must be random, for which we used stochastic modeling. In order to get randomness by definition, the variables have to be independent, that is to say, no connection or correlation between them.2 Complex systems are by all definition highly interconnected. Through these connections, the components synchronize their states and the result of this are correlations between variable, meaning the system is not random.

Non-equilibrium arises out of positive links and feedback between components. Positive feedback loops are reinforcing. They drive the whole system in one direction over time. Local level events can get amplified by positive feedback to give rise to new out of equilibrium patterns on the macro level. Bank runs are a good example of this. Starting from some initial equilibrium state on the micro level, say the bank’s balance sheet where inflow and outflow are being matched, we then get some small event.3 And this balance is broken through some loss of confidence, which then gets amplified by positive feedback where the more people that lose confidence the more other people are attracted to do likewise, and this may not stop at one bank but in fact, result in the emergence of a system’s level phenomena. Thus, we see a small local phenomenon through nonlinear interactions giving rise to a large global state of disequilibrium. This is the process of emergence and it takes place through what is called symmetry breaking.

Symmetry Breaking

Within physics and mathematics, the concept of symmetry has become central and one of the most powerful tools used to describe very fundamental phenomena in terms of symmetric transformations, such as the relationships between matter and anti-matter. The other side of this is another big idea within complex systems, that of symmetry breaking. Symmetry breaking in physics describes a phenomenon where (infinitesimally) small fluctuations acting on a system which is crossing a critical point decide the system’s fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or “noise”), the choice will appear arbitrary. This process is called symmetry “breaking,” because such transitions usually bring the system from a symmetric but disorderly state into one or more definite states. Symmetry breaking is supposed to play a major role in pattern formation.4

Traffic jams are examples of symmetry breaking. We have some traffic moving smoothly along, all at the same speed and distance, then some small event occurs such as a car needing to slow down as someone crosses the street. This creates a positive feedback loop as this small event gets amplified back. During this process, the system becomes heterogeneous. People are now moving at different speeds and distances between them. The symmetry is broken. It may return to its previous equilibrium or it may move into a whole new regime as a gridlock takes hold.
As an example of symmetry breaking within economics, we might think about the emergence of globalization. On a very high level, the industrial age system of the national economy, because it was a relatively closed system, had symmetry to it. People were producing goods in one part of the system and consuming them in another. With globalization, we get symmetry breaking of the national economy. As components emerge that connect directly into the global economy, a multinational corporation comes in and builds a mine exporting everything back out to the global market, creating a broken symmetry and dis-equilibrium.

With symmetry breaking, we get some symmetry on the micro level that is violated on the macro level, giving rise to a distinct pattern of organization. Because of the nonlinear interactions of synergies and interference, within an economic system, we get symmetry breaking and the emergence of distinct patterns of organization on different levels within the economy, and this is one way of understanding economic institutions.5 Within a linear model, there is no symmetry breaking and institutions don’t really exist. In as much as they do exist, they are typically derived from exogenous factors such as transaction costs. But with nonlinear interactions, they can be formalized as an intrinsic part of the system operating on many different levels from households to businesses, cities and whole distinct economies, all of which are non-equilibrium phenomena that have emerged out of some symmetry breaking derived from a non-linear interaction within the system and give rise to a heterogeneous macro system.

Phase transition

Symmetry breaking is part of a broader process of change that is called a phase transition. A phase transition is where some small change to a quantitative control parameter leads to a systemic transformation. So for example, the control parameter might be temperature, and at some critical point when we change it only a small amount, we will get solid ice change to a gas, which is a systemic transformation. Solids and gas have fundamentally different properties. Phase transitions are typically discontinuous. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. This discontinuous change is because there is a tipping point or threshold that has to be met before the change process will be initiated. It is not a continued change but a nonlinear abrupt change during which the system is in a state of disequilibrium.6
We could take a financial crisis as an example here. There are financial institutions that go under all the time, but there has to be a critical mass of them exhibiting stress before people will start to lose confidence, and once they have, a positive feedback loop will take hold. Every new person withdrawing money will attract others to do so also. What is happening here is that we start with some equilibrium, a stable state held in place by some negative feedback, until we reach the critical point, the tipping point where positive feedback takes over. The equilibrium or symmetry is broken and we get a rapid phase transition, the emergence of some new phenomena on the macro level, a macro level dis- equilibrium that creates some new structure.

Punctuated Equilibrium

Whereas negative feedback loops lead to an equilibrium state and a stable linear state of development, where the input and output ratio to the system stays constant leading to an incremental linear progression, real world complex systems like ecosystems and economies are a whole network of both positive and negative feedback loops operating on many different levels from the micro to the macro. The overall state that the system exhibits is a product of the balance between these two. Negative feedback is holding it in its current configuration. Positive feedback is always trying to drive it out of this equilibrium.For example, as long as I stay getting up every day and going to work, I will be able to remain in my current financial state of stable development. But when I don’t go to work and stay at home playing computer games all day, I have broken out of this negative feedback loop.
There is now a broken symmetry between what I earn and my expenses, and the longer I stay out of work the farther away I am moving from this equilibrium, possibly resulting in a regime shift as I become permanently unemployed and kicked out of my house. This model to a system’s development is called punctuated equilibrium, where periods of stable development and equilibrium, where the system is dominated by negative feedback are punctuated by periods where positive feedback becomes dominate. We get a symmetry breaking and the system moves far from its equilibrium into a phase transition as it moves into a new regime, with a new attractor and new equilibrium, and once again a new set of negative feedback loops taking over, giving us this punctuated equilibrium that is a product of an interplay between negative and positive feedback development.

Path Dependency

Whereas closed linear systems are time reversible and predictable, the development of nonlinear systems is not. Because of nonlinear feedback loops, we get sensitivity to initial conditions. We cannot predict the future and we can not go backward, meaning time only goes in one way and we find ourselves on a particular path because of what happened in the past.8 The fact that we can’t go back and change it means we get path dependency and history matters. With path dependency, because of choices made in the past, we are locked into a particular set of possible states now. An example of path dependency would be a town that is built around a factory. It makes more sense for a factory to be located a distance away from residential areas for various reasons such as pollution. However, it is often the case that the factory was built first, and the workers needed homes and amenities built close by for them. It would be far too costly to move the factory once it has already been established, even though it would better serve the community from the outskirts of town. The net result of path dependency is that we never get a clean slate on which we can choose the most efficient, optimal solutions. Our economics are instead far from optimal and thus far from equilibrium. Everything does not just smooth out to a homogeneous state, but we go on with a heterogeneous state to the economy, small pockets that are different because of the historical events that brought them to this current heterogeneous state.
All of what we have been discussing is really part of the very bigger idea of emergence, the process through which novel phenomena are created as we go from the micro to the macro or during the process of a system’s development. Emergence is a very abstract and pervasive phenomenon, but all of these models from self-organization to symmetry breaking to phase transitions and punctuated equilibrium – they are all really describing this same phenomenon of emergence in different ways due to its very high level of abstraction.

Citation

1. Puu, T. (1991). Nonlinear Economic Dynamics. Nonlinear Economic Dynamics, 1-7. doi:10.1007/978-3-642-97291-1_1 2. (2017).

2. (2017). Web.cecs.pdx.edu. Retrieved 26 May 2017, from http://web.cecs.pdx.edu/~mm/dyn-comp-edge.pdf

3. Complex adaptive systems (CAS) – Very Short Introductions. (2017). Veryshortintroductions.com. Retrieved 26 May 2017, from https://goo.gl/6AU8yr

4. Gauge Theories in Particle Physics, Volume II. (2017). Google Books. Retrieved 26 May 2017, from https://goo.gl/mcikch

5. (2017). Fernandonogueiracosta.files.wordpress.com. Retrieved 26 May 2017, from https://fernandonogueiracosta.files.wordpress.com/2015/08/berger-sebastian-the-foundations-of-non-equilibrium-economics.pdf

6. (2017). Itp.uni-frankfurt.de. Retrieved 26 May 2017, from http://itp.uni-frankfurt.de/~valenti/WS13-14/all_1314_chap6.pdf

7. Gould, Stephen Jay, & Eldredge, Niles (1977).115-151. (p.145) http://www.nileseldredge.com/pdf_files/Punctuated_Equilibria_Gould_Eldredge_1977.pdfPaleobiology

8. (2017). Www-siepr.stanford.edu. Retrieved 26 May 2017, from http://www-siepr.stanford.edu/workp/swp00011.pdf

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