Positive Feedback Economies

The objective of this book is to study the economy as a complex system and to show that the presence of self-reinforcing mechanisms in very different economic problems gives rise to common regularities and common qualitative properties. Over recent years the field of complex systems theory has mushroomed. There has been an explosion of research activity within the general area of non-linear sciences, including chaos theory, interacting particle systems, self-organized criticality models, cellular automata theory, simulated annealing learning models, stochastic approximation theory, and many others. Order, disorder, self-organization, synergetics are now notions which constitute parts of the knowledge which is increasingly being transferred from one discipline to another.

The term ‘dynamical system’ describes a system that evolves in time according to a well-defined rule. More mathematically, a dynamical system is characterized by the fact that the future values of its observable variables can be given as a function of the values of the variables at the present time. The space ∑ where such variables are defined is called the phase space. The behaviour of dynamical systems is represented in a multidimensional phase space, where the state of the system at any time is represented by a point. The dynamical system describes the change in the states with time; that is, a transformation acting on ’ is given and is called a flow. More specifically, associated with each t ∈ R (or t ∈ R⁺) there is a mapping f ^t : ∑ → ∑ such that the group property holds (that is f⁰ = Id(∑) and f ^t+s = f ^t . f ^s for all t, s). The system moves from the state x ∈ ∑ to the state f ^t x after time t. A cascade differs from a flow in that the maps f ^t are defined only for integer t. The evolution of a system starting from a given initial state — that is, from a given point x in the phase space — is represented by a trajectory in this space; that is, the trajectory of x is the set {f^tx}. If t > 0 we use the prefix semi- for flows, trajectories, etc. If f^tx = x for all t, then x is an equilibrium point. If f^t+Tx=f^tx for all t and for some T ≠ 0, then the trajectory { ^t x} is said to be periodic. Periodic trajectories are closed and are often called cycles. If a set A ⊂ ∑ is such that f ^t A=A for all t, then A is said to be an invariant set.

The possibility that agents may react discontinuously to continuous changes in their environments does not seem to have been sufficiently investigated in economics. Intuition suggests that continuously changing causes should produce continuous effects. However, as we discussed in Chapter 2, catastrophe theory makes clear that the occurrence of discontinuities in smoothly evolving systems is not an unlikely event. One of the purposes of this chapter is to show that ‘catastrophes’ can arise in a simple model of the adoption of innovations when self-reinforcing mechanisms are introduced.

Path-dependent dynamic systems with self-reinforcing mechanisms often have a multiplicity of possible asymptotic states. The initial state and early random events push the dynamics into the domain of one of these asymptotic states and thus select the structure that the system eventually locks into. This issue has been tackled in several economic examples, by Arthur, David and others, dealing with sequential choices between competing technologies when increasing returns to adoption are present. These models have shown that such systems display the properties of multiple equilibria, possible inefficiency, lock-in, path-dependence and symmetry-breaking which we introduced in Chapter 1, section 1.2. A question arising in this context is the following: if an economic system is locked-in to an inferior local equilibrium, is ‘escape’ into a superior one possible? Do we need policies for the economic system, or will spontaneous actions at a local level suffice?

It has long been argued that as a result of externalities the value of choosing a technology may be enhanced by the fact that other firms have previously chosen it. This issue has been tackled by Brian Arthur and Paul David in their writings on cumulative causation occurring in path-dependent processes which have affected much of the literature about the dynamics of allocation under increasing returns occasioned by learning-by-doing and learning-by-using phenomena. The idea is that ‘history matters’ when increasing returns to adoption are introduced. If one technology gets ahead by good fortune it gains an advantage, with the result that the adoption market may ‘tip’ in its favour and may end up dominated by it. With different circumstances, a different technology might have been favoured early on and it might have come to dominate the market. Thus in competition between technologies with increasing returns there are possible multiple equilibria. As to which actual outcome is selected from these multiple candidate outcomes, it is argued that the prevailing outcome turns out to depend on the path which has been initially chosen. In particular, the resulting outcome may be inefficient; that is, the market may be locked-in to the ‘wrong’ technology.

There are situations in which agents' behaviour or agents' choices depend on the behaviour or the choices of other agents. In these cases we have to look at the system of interactions between individuals and their environment. Schelling (1978) provides examples of such interactive behaviour. People distribute themselves and congregate at parties and receptions, or form crowds at a rally, a riot or a spectacle, without following a single mode of behaviour. Sometimes people want to be close, sometimes spread out; the people on the edge of a crowd may be pushing to get in and the people in the middle are being crushed. The best and commonest examples are from everyday life. People get separated and integrated by sex, race, age, language, social status or by patterns of acquaintance and friendship. Age at marriage and age differences between spouses are affected by the ages at which others marry. Divorce and the prospects of remarriage depend on whether there is a high rate of turnover in particular age brackets. What other people in the same area are doing heavily influences other kind of behaviours such as the choice of the language, the diffusion of rumour, gossip and news, information and misinformation. The same kind of factor explains the formation of mobs and riots, panic behaviour, rules of the road, taste, style and fashion.

There has been a steady development of dynamic analysis in economics in recent decades, both in theoretical work and in empirical implementations. Although the theoretical equations were frequently nonlinear, the empirical methods often employed the tools of linear stochastic analysis; moreover, the economic theory underlying dynamical systems very often tended to emphasize amplitude-reducing behaviour, that is, negative feedbacks. In this book we have taken the perspective of positive feedback economies and have developed models where the observed distribution of economic activity might be determined by history.