Markets, Productivity, and Happiness in a Historical Perspective

Stability issues in large-scale dynamic systems : from Goodwin’s ”dynamical coupling” to May’s instability conjecture

Raybaut Alain, Gredeg/ CNRS/ Université Cote d'Azur

This contribution focusses on some key advances developed in the 1950s and 1970s on stability issues in large-scale economic systems. These systems refer to large dimensional dynamical systems with interacting elements. This excludes the standard literature on stability analysis of a competitive general equilibrium. One of the major questions is then to determine whether increasing complexity (size and structure of interaction) tends to stabilize or destabilize the system. The initial treatment of the problem can be found in Goodwin (1947) with the explicit introdduction the concept of dynamical coupling in economics. It also discusses how some formal difficulties may be reduced to much simpler configurations. At the same time, the issue was also directly addressed by Hawkins (1948). Instablility is usually attributed to failures of market mechanisms. For Hawkins, deeper sources of instabilities lie in the coupling of any parts of an economy. The article gives sufficient conditions of stability in terms of the degree of coupling between different branches of the economy in a simple linear model. The question directly refers to the set of articles dedicated in the 1950s to the ”matrix multiplier” approach and input-output analysis. In this perspective, Solow (1951, 1952) investigated the formal similarity between the multi-sector multiplier, input-output models and coupling in dynamic linear models, insisting on the importance of the internal structure of the models with a similar approach to that of Debreu and Herstein in their contribution on square nonnegative matrices. Second, he showed how macroeconomic stability depends on the way stable and unstable sectors are coupled by money or commodity flows. The issue was reformulated in the 1970s independently of the of input output analysis in an interdisciplinary perspective. In his seminal contribution, ”Will a large complex system be stable ?”, May (1972) reconsiders the widely accepted assumption in theoretical ecology that large and highly connected systems would be more stable than simple (smaller) ones less intricately connected. May states clearly what formal conditions are needed for complex systems to be stable, and shows that in general increasing the size or adding connections to the interaction network will favor instability. Quite a few authors have also been concerned with stability analysis of large scale systems with applications to competitive-cooperative interactions among species,, agents, industries or countries. In this perspective, the concept of connective stability was introduced for functional nonlinear differential systems. Finally, let mention the recent contribution by Bouchaud and Moran (2019) building directly on the ideas of Hawkins and May showing that large economic systems are likely to be unstable, as also anticipated in the 1990s by Bak, Chen, Scheinkman, and Woodford in the context of self organized criticallity.

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Keywords: Dynamical coupling, Instability, Stability, Large- Scale dynamical systems, Connective stability,