# Nonlinear Systems

# Nonlinear Systems

All real world systems are nonlinear: Straight lines cannot, in practice, go on forever, nor can forces and interactions in natural or social systems. Nonlinearity arises in real economic systems both from human behavior—for example, wage demands varying as a function of the rate of employ-ment—and from the interaction of economic variables—for example, the multiplication of wage rates by the number of workers to calculate the wage bill. Any model that omits these nonlinearities, either by assuming linear behavioral functions or by assuming that a variable remains constant in order to avoid interactive nonlinearities, necessarily reduces its capacity to model the actual economy.

Idealized linear systems can be hypothesized, and the mathematical analysis of these is long established, well-known, and generally results in closed-form symbolic solutions in which the system state at any point in time is a function of the system’s parameters. Moreover, in the vicinity of an equilibrium, the linear component of a system, which can be extracted from a mathematical model by a polynomial expansion, dominates the nonlinear components. Thus if a system is stable about an equilibrium, or can be constrained to remain in the vicinity of an equilibrium, its dynamics can be modeled using linear methods.

The mathematical analysis of nonlinear systems, on the other hand, is a recent development, and in general does not result in symbolic solutions. Instead, a nonlinear system must be numerically simulated, and in the subsets classed as either chaotic or structurally unstable, the time path of a system depends upon its initial conditions. Scientists in general therefore had a strong incentive to remain in the linear realm.

Since many real world systems did not meet the conditions for linear analysis, nonlinear analytic techniques were gradually developed, leading to what was initially called chaos theory and is now known as complexity theory. Nonlinear methods play a major role in most sciences today, but their uptake in economics has been noticeably more limited.

## NONLINEAR ECONOMICS

Nonlinear economics began in the 1940s, when Hungarian economist Nicholas Kaldor (1908–1986) made the prescient observation that a model with linear ex-ante investment and savings functions could not explain the trade cycle. With savings and investment modeled as linear functions of employment, if the savings function were the steeper of the two, the model displayed “*more* stability than the real world appears, in fact, to possess.” On the other hand, if the investment function were steeper, then the system “would always be rushing either towards a state of hyper-inflation with full employment, or towards a state of complete collapse with zero employment, with no resting-place in between” (Kaldor 1940, p. 80). Since neither result could be justified, Kaldor surmised that “we are left with the conclusion that the *I* (*x* ) and *S* (*x* ) functions cannot both be linear, at any rate over the entire range” (Kaldor 1940, p. 81).

The adoption of nonlinear methods in economics after this insight was very limited, and Kaldor later argued that this was because economics took for granted “that the economy always approaches, or is near to, a state of equilibrium” (Kaldor 1972, p. 1239). Economists therefore tended to rely upon comparative statics methods, even when the relevant linear model was unstable under reasonable parameter values—as in the case of the linear model of supply and demand, which is unstable under the realistic condition that the price elasticity of supply exceeds that of demand.

While Kaldor pointed out the need for nonlinear analysis in economics, the main pioneer of nonlinear models in economics was Richard Goodwin (1913–1996). Drawing his inspiration from the French mathematicians Henri Poincaré and Philippe Le Corbeiller (b. 1911), Goodwin developed many nonlinear models, with his signature contribution implementing Karl Marx’s class-struggle cycle model as a predator-prey system—in which technically, the capitalists were the prey and workers the predators.

There is now a substantial research tradition in nonlinear dynamics within economics that overlaps with the application of chaos, complexity, and evolutionary theories, and fractal analysis to economics. There are numerous nonlinear dynamical models of macroeconomics, microeconomics, and finance market phenomena, including the aggregate business cycle, the individual market cobweb cycle, and stock market crashes. Many econometric methods to test economic time series for nonlinear data structures (e.g., the Hurst exponent, the Brock–Dechert–Scheinkman (BDS) statistic, Smooth Transition Auto-Regression) exist, though their robustness at determining whether nonlinear causal structures exist in noisy linear time series is limited.

There are journals devoted to nonlinear economic analysis—such as *Studies in Nonlinear Dynamics & Econometrics*, and *Nonlinear Dynamics, Psychology, and Life Sciences—* as well as others where nonlinear analysis features frequently—such as the *Journal of Economic Dynamics and Control*, *Macroeconomic Dynamics*, the *Review of Economic Dynamics,* and *Structural Change and Economic Dynamics*. Most contributions from the developing field of econo-physics could be characterized as essentially nonlinear, including those applying nonextensive statistical mechanics to the analysis of financial market data.

Despite this flowering, nonlinear analysis in economics is hampered by two dilemmas. The first, peculiar to economics, is that the dominant pedagogic and research tendency in economics is to model economic processes as equilibrium phenomena. The second, generic problem is best captured by John von Neumann’s (1903–1957) apocryphal aphorism that a general theory of nonlinear phenomena is akin to “a theory of non-elephants”: While linear analysis is well defined, the variety of nonlinear phenomena is so enormous that it bedevils systematic analysis (Bak and Paczusku 1995, p. 6690).

**SEE ALSO** *Linear Systems*

## BIBLIOGRAPHY

Asada, Toichiro, Pu Chen, Carl Chiarella, and Peter Flaschel. 2006. Keynesian Dynamics and the Wage-Price Spiral: A Baseline Disequilibrium Model. *Journal of Macroeconomics* 28 (1): 90–130.

Bak, Per, and Maya Paczuski. 1995. Complexity, Contingency, and Criticality. *Proceedings of the National Academy of Sciences of the United States of America* 92 (15): 6689–6696.

Blatt, John M. 1983. *Dynamic Economic Systems: A Post Keynesian Approach*. Armonk, NY: M. E. Sharpe.

Holt, Matthew T., and Lee A. Craig. 2006. Nonlinear Dynamics and Structural Change in the U.S. Hog-Corn Cycle: A Time-Varying STAR Approach. *American Journal of Agricultural Economics* 88 (1): 215–233.

Goodwin, Richard M. 1951. The Nonlinear Accelerator and the Persistence of Business Cycles. *Econometrica* 19 (1): 1–17.

Goodwin, Richard M. 1967. A Growth Cycle. In *Socialism, Capitalism and Economic Growth*, ed. C. H. Feinstein, 54–58. Cambridge, U.K.: Cambridge University Press.

Kaldor, Nicholas. 1934. A Classificatory Note on the Determinateness of Equilibrium. *Review of Economic Studies* 1 (2): 122–136.

Kaldor, Nicholas. 1940. A Model of the Trade Cycle. *Economic Journal* 50 (197): 78–92.

Kaldor, Nicholas. 1972. The Irrelevance of Equilibrium Economics. *Economic Journal* 82 (328): 1237–1255.

Kyrtsou, Catherine, and Apostolos Serletis. 2006. Univariate Tests for Nonlinear Structure. *Journal of Macroeconomics* 28 (1): 154–168.

Marx, Karl. *Capital I*. Marxists Internet Archives. http://www.marxists.org/archive/marx/works/1867-c1/.

Rose, Hugh. 1967. On the Non-Linear Theory of the Employment Cycle. *Review of Economic Studies* 34 (2): 153–173.

Rosser, J. Barkley. 2000. *Mathematics, Microeconomics and Finance*. Vol. I of *From Catastrophe to Chaos: A General Theory of Economic Discontinuities*. New York: Springer.

Zhou, Wei-Xing, and Didier Sornette. 2003. Evidence of a Worldwide Stock Market Log-Periodic Anti-Bubble since Mid-2000. *Physica A* 330 (3–4): 543–583.

*Steve Keen*

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