# Economic applications of dynamical systems theory.

1. IntroductionOn Schenk-Hoppe's reading, dynamic economic modeling comprises optimizing behavior of agents and temporary equilibrium conditions. The perturbation is an empirical fact that many randomly fluctuating variables are nonstationary. Janssen et al. clarify the relation connecting the choice between two alternative projects and the substitutive financial operations, which are obtained formally as difference operations between two operations (of investment or financing).

2. Computable Economics and Dynamic Complexity

Benson states that model-builders who advocate policy based on their mathematical manipulations are similar to empirical policy analysts (they generally do not want their normatively-driven policy prescriptions to be attacked) and that individuals and groups have two principal ways of augmenting their wealth, voluntary cooperation in joint production and trade, and taking wealth from others by force or guile. Benson contends that recognition of both kinds of incentives, to cooperate voluntarily and to transfer through coercion, enhance our understanding of institutions and their behavioral implications. (1) Barkley Rosser examines the rising competition between computational and dynamic conceptualizations of complexity in economics: computable economics views the complexity as something rigorously defined based on concepts from probability, information, and computability criteria; dynamic complexity is based on whether a system endogenously and deterministically generates erratically dynamic behavior of certain kinds. Dynamic complexity and such concepts as emergence are useful for understanding economic phenomena. (2) Donnelly and Embrechts believe that there should be a reliance on sophisticated mathematics. The root of the Crisis was the transfer of the risk of mortgage default from mortgage lenders to the financial market at large. Many market participants either did not realize this was happening or did not think that it was significant. The coupon payments received by the holders of the CDO tranches depend directly on the defaults occurring in the underlying portfolio of assets. Donnelly and Embrechts believe that it is imperative that the financial world considers what the model they use implies about frequency and severity of extreme events. Models based on the normal distribution should be used in conjunction with several different models, some of which should adequately capture extreme events. Donnelly and Embrechts explain how AIG came close to bankruptcy and draw some relevant lessons from their risk management failures. (3)

3. The Role of Dynamical Systems Theory in Economics

van Daal and Jolink deal with mathematics in economics, free competition, and price determination under a regime of free competition and utility (elements which are basic in Walras's theory of pure economics). According to Walras, in discussing the application of mathematics we should distinguish between application to economic theory and application to economic practice. In the case of pure economic theory we study and explain facts (the use of mathematics is motivated by the need for a method of analysis). In economic practice mathematics is used to evaluate certain consequences of an economic measure. van Daal and Jolink stress that in most cases Walras's mathematics should be regarded as a method of analysis in abstract terms, rather than as a method of calculation. Mathematics could be of practical relevance by formulating general rules. (4) Schenk-Hoppe discusses the role of dynamical systems theory in economics, insisting on the potential impact and usefulness of the theory of random dynamical systems for economic modeling and economic analysis. Dynamic economic models are, in general, not related to dynamical systems theory. According to Schenk-Hoppe, the derivation of a dynamical systems description of an economic model requires that (1) a state space can be defined, (2) all implicit equations in the economic model have solutions for different initial states and variables, and (3) these solutions exhibit a suitable time-structure in the dependence on states and variables. Schenk-Hoppe surveys recent progress in the application of random dynamical systems theory in stochastic economic growth, discussing a descriptive growth model in which the decision of households is not modeled, an overlapping generations model in which households live for two periods, and the optimal growth model with an infinitely-lived representative agent. The assumption on the consumption behavior of households is compatible with the assumption that capital is irreversible. (5) Janssen et al. cover financial mathematics in a deterministic context, assuming that the monetary income and outcome movements will happen and in the prefixed amount. Janssen et al. formalize the indifference curves and the principle of financial indifference in objective terms, defining financial factors, rates and intensities for lending and discounting operations, in relation to the possible distribution of interest payments in the deferment period. In relation to a financial it is important to investigate the existence and the properties of regimes which are decomposable and uniform. (6)

4. Economic Growth and Dynamical Systems

Haddad and Chellaboina are more interested in the future behavior of the system as opposed to the past. A dynamical system defines a continuous flow in the state space. Haddad and Chellaboina outline the salient changes for time-varying dynamical systems as compared to time-invariant dynamical systems, and state the key results on the existence and uniqueness of solutions for time-varying dynamical systems needed for later developments. Constructing the actual domain of attraction of a nonlinear dynamical system is system trajectory dependent. There exists a continuously differentiable time-independent Lyapunov function for asymptotically stable, time-invariant nonlinear dynamical systems. It is not always necessary to construct a time-varying Lyapunov function to show stability for a nonlinear time-varying dynamical system. For time-invariant dynamical systems the invariance principle can be used to relax the strict negative-definiteness condition on the Lyapunov derivative. Haddad and Chellaboina consider open dynamical systems wherein the system interaction with the environment is explicitly taken into account through the system inputs and outputs. Lagrangian and Hamiltonian dynamical systems can be formulated as special cases of dissipative dynamical system theory. Haddad and Chellaboina present sufficient conditions for an absolute stability problem involving a dynamical system with memoryless, time-varying feedback nonlinearities, and construct Lyapunov functions for interconnected dynamical systems by appropriately combining storage functions for each subsystem. (7) On Zhang's reading, the theory of differential equations has become an essential tool of economic analysis. The growth rate is affected by many factors, such as the current state of the economic system, accumulated knowledge of the economy, international environment etc. Zhang introduces some economic mechanisms to avoid the decline of living standard, examines a dynamic model to see how leisure time and work hours change over time in association with economic growth, and examines dynamics of sexual division of labor and consumption in association of modern economic growth, illustrating increases of women labor participation as a "consequence" of economic growth as well as changes of labor market conditions. Zhang examines dynamic properties of some frequently-applied economic models, such as the competitive equilibrium model, the Walrasian-Marshallian adjustment process, the Tobin-Blanchard model, and the Ramsey model. Zhang introduces concepts, theorems, and methods in differential equation theory which are widely used in contemporary economic analysis, and is concerned with how differential equations can be applied to solve and provide insights into economic dynamics. (8) In Foster's view, dissipative structures such as firms cannot enter new niches or escape the stationary states to which they tend through optimization exercises alone. What is decisive in complex systems is selection. Optimization will depend upon the opportunities and threats that systems face. Completely connected networks (fields) can never exist across an economy. (9)

5. Conclusions

Haddad and Chellaboina view a dynamical system as a precise mathematical object defined on a time set as a mapping between vector spaces satisfying a set of axioms. Advances in feedback control theory have been intricately coupled to progress in dynamical system theory. Advances in Lyapunov-based methods have been developed for analysis and control design for numerous classes of nonlinear dynamical systems. Zhang deals with nonlinear planar differential equations, carrying out local analysis, providing conditions for validity of linearization and relations between linear systems and almost linear systems with regard to dynamic qualitative properties.

REFERENCES

(1.) Benson, B.L. (2009), "Economic Dissociative Identity Disorder: The Math Gamer, the Anti-Policy Econometrician and the Narrative Political Economist", Econ Journal Watch 6(3): 364-373.

(2.) Barkley Rosser, J., Jr. (2006), "Computational and Dynamic Complexity in Economics," RWP, June.

(3.) Donnelly, C. and Embrechts, P. (2010), "The Devil is in the Tails: Actuarial Mathematics and the Subprime Mortgage Crisis", ASTIN Bulletin 40(1): 117-149.

(4.) van Daal, J. and Jolink, A. (1993), The Equilibrium Economics of Leon Walras. New York: Routledge, 3-7.

(5.) Schenk-Hoppe, K.R. (2000), "Random Dynamical Systems in Economics," WP-67, Institute for Empirical Research in Economics, University of Zurich, December.

(6.) Janssen, J. et al. (2009), Mathematical Finance. Deterministic and Stochastic Models. London: ISTE and Hoboken: John Wiley, 12-147.

(7.) Haddad, W.M. and Chellaboina, V. (2008), Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton, NJ: Princeton University Press, 5497.

(8.) Zhang, W.-B. (2005), Differential Equations, Bifurcations, and Chaos in Economics. New Jersey: World Scientific, 5-240.

(9.) Foster, J. (2004), "From Simplistic to Complex Systems in Economics," DP335, October, School of Economics, University of Queensland.

DUMITRU BALA

dumitru.bala@ucv.ro

University of Craiova

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Title Annotation: | Proceedings of the 5th World Congress on the Advancement of Scholarly Research in Science, Economics, Law, and Culture: May 27-30, 2010 New York |
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Author: | Bala, Dumitru |

Publication: | Economics, Management, and Financial Markets |

Article Type: | Report |

Geographic Code: | 4EXRO |

Date: | Jun 1, 2011 |

Words: | 1496 |

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