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## Über dieses Buch

This volume constitutes the Proceedings of the "Nonlinear Dynamics in Economics and Social Sciences" Meeting held at the Certosa di Pontignano, Siena, on May 27-30, 1991. The Meeting was organized by the National Group "Modelli Nonlineari in Economia e Dinamiche Complesse" of the Italian Ministery of University and SCientific Research, M.U.RS.T. The aim of the Conference, which followed a previous analogous initiative taking place in the very same Certosa, on January 1988*, was the one of offering a come together opportunity to economists interested in a new mathematical approach to the modelling of economical processes, through the use of more advanced analytical techniques, and mathematicians acting in the field of global dynamical systems theory and applications. A basiC underlying idea drove the organizers: the necessity of fOCUSing on the use that recent methods and results, as those commonly referred to the overpopularized label of "Chaotic Dynamics", did find in the social sciences domain; and thus to check their actual relevance in the research program of modelling economic phenomena, in order to individuate and stress promising perspectives, as well as to curb excessive hopes and criticize not infrequent cases where research reduces to mechanical, ad hoc, applications of "a la mode" techniques. In a word we felt the need of looking about the state of the arts in non-linear systems theory applications to economics and social processes: hence the title of the workshop and the volume.

## Inhaltsverzeichnis

### Economics and the Environment: Global Erodynamic Models

Abstract
Simple dynamical systems theory evolved from celestial mechanics in the work of Poincaré a century ago. Complex dynamical systems theory (also known as systems dynamics) began during World War II, with the work of Von Bertalanffy on general systems theory and Wiener on cybernetics. Cellular dynamical systems theory developed in the early days of biological morphogenesis in the work of Rashevsky and Turing. Since the advent of massively parallel computation, these modeling strategies have been increasingly used to simulate highly complex natural systems. The challenge to understand our global problems — combining physical systems of the atmosphere and ocean (Chaos) with biological systems of the biosphere (Gaia) and the social systems of human and other species (Eros) — will test and extend our mathematical and scientific capabilities. The name erodynamics has been coined to describe this application of dynamical systems theory to the complex global system of our human civilization and environment. In this minicourse we develop the basic concepts of erodynamics in the frame of economics and the environment.
Ralph Abraham

### Complexity in Deterministic, Nonlinear Business-Cycle Models — Foundations, Empirical Evidence, and Predictability

Abstract
Business-cycle theory represents one of the oldest fields in economics. While it was treated as a nearly esoteric field in specialized graduate texts during the late 1960s and early 1970s, the last fifteen years saw it resurrecting even as a synonym for dynamic macroeconomics. The Rational Expectations literature of the late 1970s and early 1980s and the development of sophisticated econometric tools in investigations of an economy’s fluctuations occasionally seemed to encourage the believe that business-cycle theory was an invention of the so-called New Classical economics. However, it is a fact that the observed cycling of an economy constituted the major impetus for many a classical and neoclassical economist in the 19th and early 20th century to engage in economic theorizing at all. Haberler’s (1937) seminal text on the history of business-cycle theory demonstrates in an enlightening fashion that the ups and downs in economic activity were central not only in - to name just a few - Hawtrey’s (1913), Hayek’s (1933), Marx’s (1867), Pigou’s (1929), or Sismondi’s (1837) work but that numerous, usually forgotten writers concentrated on oscillations in particular markets or the entire economy.
Hans-Walter Lorenz

### Fractal “box-within-a-box” Bifurcation Structure

Abstract
The “box-within-a-box” (or “embedded boxes”) bifurcation structure is a typical fractal arrangement of a set of bifurcation points located in the parameter space of a given dynamic system. It is recalled that a bifurcation is a qualitative change of this system under the effect of variation of its parameters.
Christian Mira

### Recurrence in Keynesian Macroeconomic Models

Abstract
The development of new mathematical techniques in the theory of nonlinear dynamic systems has greatly enlarged the analytical basis for modern business cycle theory in economics. The elaborate theory of one-dimensional systems offers a large spectrum of methods and results indicating a great degree of different dynamic phenomenae which were not known before. Its impact on dynamic economics can already be found in a large number of recent publications. One of the most surprising class of results is given by the long list of contributions examining chaotic behavior in one-dimensional competitive systems (see for example the contributions in [10]).
Volker Böhm

### Economic Nonlinear Dynamic Development

Abstract
The conception of a limit cycle was formulated towards the end of the last century. A practical, usable form was first developed by van der Pol and widely used in the physical sciences. The first qualitative model in economics was proposed by Kalecki in the early thirties but, being linear, was seriously criticized by Frish on the grounds that it was structurally unstable. Then came the dramatic impact of the Keynes General Theory, which led to an altered perception of the problem.
Richard M. Goodwin

### Random Walk vs. Chaotic Dynamics in Financial Economics

Abstract
During the past three decades, financial economists have studied in detail the behavior of stock market prices and the prices of derivative securities issued on such stocks. By price behavior we mean the dynamic, period by period, change in the price level of a given stock, such as, the daily closing price of an IBM share and the daily settlement price of the corresponding put or call. The extensive literature that addresses these twin problems is known as the market efficiency theory and the option pricing theory. Both theories constitute significant pillars of modern financial economics and despite various puzzles and anomalies, such as the October 1987 stock market crash, that cannot be explained by market efficiency, there are currently no competing theories that are widely accepted.
A. G. Malliaris, G. Philippatos

### A New Keynesian Model of the Business Cycle and Financial Fragility

Abstract
Two competing procedures are currently adopted in modelling the cyclical dynamics of output. According to the first one, output movements must be conceived of as responses of a linear system to stochastic disturbances (Frisch, 1933; Slutstky, 1937; Lucas, 1975). Alternatively, the second one interprets the oscillating path of income as a built-in feature of a non-linear deterministic system (Kaldor, 1940; Hicks, 1950; Goodwin, 1982; Grandmont, 1985). According to us, there is no reason why these two explanations should not co-exist: in this paper output dynamics is determined by a non-linear system of difference equations which takes into account also stochastic shocks (similar attempts in this direction have been recently carried out by Greenwald and Stiglitz, 1988a; Day and Lin, 1991; Delli Gatti et al., 1992). In our model the basic ingredients of both explanations are incorporated in the investment equation. Infact the propensity to invest is modelled as a non-linear function of income affected by a stochastic disturbance. Stochastic disturbances enter the process of income determination also through the capital assets price equation. Therefore the dynamics of the main macro-variables is jointly determined by the deterministic non-linear difference equations system and the random process which generates “clouds” around a deterministic cycle (when the shock is additive), and “jumps” from a cycle to another (when the shock affects the value of the parameters).
Domenico Delli Gatti, Mauro Gallegati

### Non-Walrasian Equilibria in a Labour-Managed Economy

Abstract
This paper studies the existence and stability of non-Walrasian equilibria in a labour-managed economy. The main finding is that non-Walrasian equilibria exist and constitute a one-dimensional smooth manifold; they are stable but not asymptotically stable.
Angela De Sanctis, Gerd Weinrich

### Endogenous Cycles, Increasing Returns and Global Bifurcations in an Imperfectly Competitive Economy

Abstract
The purpose of this paper is firstly to analyze the possibility and secondly to characterize the consequences of aggregate externalities of increasing returns to scale in a decentralized growth model specialized to incorporate aggregate externalities of stock which are intended to stylize in a convenient way natural limits on expansion on the economy.
Jean Pierre Drugeon

### The Dynamics of Real Wages

Abstract
Within the subject of labour market dynamics, controversies about the relationship between employment and real wages over the business cycle have a long history. In this perspective they raise theoretical and empirical issues, as is witnessed by the lack of well established “stylized facts.” To the traditional dichotomy between those who believe in a countercyclical movement typically ascribed to Keynes (1936) and the supporters of the procyclical pattern (among others, Keynes (1939) himself),2 a new attitude is emerging that believes this pattern to be very complex. The complexity derives either from the influence of stochastic elements or from the presence of factors that play different roles in different historical contexts. As Zarnowitz (1985, p. 543) points out: “The evidence is mixed and not conclusive. It varies with the choice of the deflator, the characteristics of the period covered, methods and dates.”
Piero Ferri, Edward Greenberg

### Oligopolistic Competition; from Stability to Chaos

Abstract
This note deals with the classic Cournot (1838) model of oligopolistic competition, thoroughly reviewed in Tirole (1988). Under discussion is an industry composed of finitely many firms i ∈ I, all producing the same homogeneous good for one competitive market. Firm i furnishes the quantity qi ≥ 0 at cost ci(qi ), thus obtaining the profit
$${{\pi }_{i}} = p(Q){{q}_{i}} - {{c}_{i}}({{q}_{i}}).$$
Sjur D. Flåm

### Nearest Neighbor Forecasts of Precious Metal Rates of Return

Abstract
In 1953 Kendall presented the results of a statistical study intended to separate out regular price cycles in the time-series of certain assets. Surprisingly he was not able to find such patterns. Kendall’s discovery had been anticipated by Bachelier, Working and probably others. The observation that asset prices appear to behave randomly stimulated a great deal of empirical and theoretical research. Kendall expected opposition to his findings by economists. However under the label of the “efficient markets hypothesis,” it is now well understood that such randomness is not evidence of irrationality in the market processes. As a result, such terms as martingale, random walk and fair game have become standard textbook material in finance. For a classic presentation of the development of these ideas see Fama (1970, 1991).
Murray Frank, Thanasis Stengos

### On a model of financial crisis: critical curves as new tools of global analysis

Abstract
Many models applicable to several fields are mathematically described by a nonlinear system of two difference equations, or map of the plane ℝ2 into itself. This is particularly true in the economic context, where the variables often change at discrete times by their own nature or definition. Generally, in these models, the nonlinearities are such that the resulting map is one with a non-unique inverse, that is, an endomorphism. Examples Can be found in [1–4]. The model described in [3] is a particular case of the more general one presented in [4]. They interpret “economic cycles” and “financial crisis” endogenously generated from the nonlinear interaction between the “goods market” and the “money market”. We use this model to illustrate how new analytical tools, the critical curves, can be used to study (in endomorphisms) the local-global attractivity of fixed points, invariant curves, cycles or of other attracting sets (regular and chaotic), as well as to determine and characterize global bifurcations which cause changes in the structure of invariant sets, or in the structure of trajectories.
Laura Gardini

### Phase-Locking in a Goodwin Model

Abstract
Mode-locking is a phenomenon that arise in nonlinear dynamical systems, when two or more cyclic motions interact with one-another. In the first place mode-locking means that the oscillations adjust to each other so as to get in step. However, the adjustment is more complicated than this because each mode contains harmonics and subharmonics which may also try to get in step. Many economic phenomena can be thought of as coupled oscillators. Examples are the interaction between seasonal fluctuations and the business cycle, the interaction between different sectors in an economy (Lorenz, 1987a), and the interaction between different national or regional economies (Lorenz, 1987b).
Christian Haxholdt, Erik Reimer Larsen, Mich Tvede, Erik Mosekilde

### Complexity of Optimal Paths in Strongly Concave Problems

Abstract
Several economic problems are often formulated in terms of a dynamic optimization model. Although a broad variety of mathematical models have been studied, the optimal capital accumulation model originated in Ramsey (1928) has played a prominent role. From the mathematical point of view, it can be formulated as a discrete-time, infinite-horizon concave optimization problem:
$$\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{{W}_{\delta }}({{x}_{0}}) = {{{\sup }}_{{({{x}_{t}})}}}\sum _{{t = 0}}^{\infty }V({{x}_{t}},{{x}_{{t + 1}}}){{\delta }^{t}}} & {s.t.} \\ \end{array} } \hfill \\ {({{x}_{t}},{{x}_{{t + 1}}}) \in D and {{x}_{0}} is fixed,} \hfill \\ \end{array}$$
(P)
where V:D → R is a concave function defined over a convex and closed set D ⊂ X × X, and the initial condition x 0 belongs to X = pr1 (D) ⊂ Rn.
Luigi Montrucchio

### Acyclicity of Optimal Paths

Abstract
It is well known that the stability of optimal paths for infinite horizon concave problems is not assured for all discount factors. Along the ideas developed in Boldrin and Montrucchio (IER, 1988), we provide new results of stability which are related to the notion of acyclicity. Some order relations are introduced which can be seen as generalizations of Liapunov theory. Our results are quite complete for one-dimensional case. In higher dimensions we state some new results.
Luigi Montrucchio, Nicola Persico

### Expected and Unexpected Distributive Shocks: An Analysis of Short and Long Run Effects

Abstract
The theory of economic growth has faced the problem of shocks since its very beginning: a shock is a sudden change of one or several relevant parameters of the economic system which perturbs the growth path and sometimes significantly changes its direction.
Franco Nardini

### Nonstandard General Equilibrium

Abstract
When writing his book Walras (1874–1877) had no mathematical tools, actually the so called fixed points theorems, to prove the existence of economically meaningful solutions to his one period competitive general equilibrium models, from now on shortly called walrasian models or abstract models. But he felt reasonably sure that (substantially) he had solved the existence problem in two ways, distinct both conceptually and practically. The first way was by counting the number of (independent) equations forming the model and the number of unknowns, to show that these numbers are equal. The second way discovered by Walras, a more practical one, was to make a conceptual experiment consisting in assuming that there is a particular agent, called by Walras auctioneer, who chooses “at randorn” one positive price vector and then records market demand and market supply of every good. In a second step, once ordered all markets in some sequence, the auctioneer rises prices in markets in which the excess demand is positive and decreases prices in those markets for which excess demand is negative. Walras felt sure that in the end all excess demands would be reduced (in absolute value) to zero, so obtaining a vector of equilibrium prices for his one-period models. During the whole process of manipulating prices no agent is allowed to produce or to consume: economic operators must only inform the auctioneer about their supplies and demands at the various prices, as if those prices where equilibrium ones; so `tâtonnements’ do not work in calendar time: time is here merely logical time. Moreover, to introduce an auctioneer means to assume that walrasian general equilibrium has a lot of structure.
Pier-Carlo Nicola

### Real Indeterminacy, Taxes and Outside Money in Incomplete Financial Market Economies: I. The Case of Lump Sum Taxes

Abstract
The basic model of exchange economies with incomplete financial markets and nominal assets has been thoroughly studied. Cass (1984), Werner (1985), Duffie (1987) and others have proved existence of equilibria. Balasko and Cass (1989) and Geanakoplos and Mas-Colell (1989) have shown that, generically, the equilibrium allocations exhibit a degree of indeterminacy related to the number of states of the world and the number of available financial instruments.
Antonio Villanacci

### Backmatter

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