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Über dieses Buch

This book was first pUblished in 1989 as volume 336 in the Springer series "Lecture Notes in Economics and Mathematical Systems", and it reappeared in a 2nd edition as a Springer monograph in 1991. After considerable revisions it appeared in a 3rd edition in 1993. The origin, still visible in the 3rd edition, was the joint work of the author with Professor Martin J. Beckmann, documented in two co-authored mono­ graphs "Spatial Economics" (North-Holland 1985), and "Spatial Structures" (Springer-Verlag 1990). Essential dynamics had, however, been almost com­ pletely lacking in these works, and the urge to focus the dynamic issues was great. To fill this particular gap was the aim of the previous editions, and so the spatial aspect provided core and focus. In the present edition a substantial quantity of spatial issues have been removed: All those that were dynamic only in the sense that structures were characterized which were structurally stable, or robust in a changing world. The removed material has meanwhile been published as a separate mono­ graph under the title "Mathematical Location and Land Use Theory" (Springer-Verlag 1996).

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
Dynamic analysis in economics is as old as economics itself. A glance at the subject index in Joseph Schumpeter’s “History of Economic Analysis” from 1954 is sufficient to convince you about this. Even dynamic mathematical models are fairly old. The cobweb model of price adjustments for instance dates back to 1887.
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2. Differential Equations

Abstract
There is no more useful tool for the study of differential equations, in particular if they are in two dimensions, than the phase portrait. Many important systems both in physics and in economics in fact live in two dimensions. All second order systems are two dimensional. To this category belong all the oscillators, exemplified by the mathematical pendulum, or by the Samuelson-Hicks business cycle model if put in continuous time. It should be remembered that a second order differential equation, as characteristic of an oscillator, can always be put in the style of two coupled first order equations.
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3. Iterated Maps

Abstract
In last chapter we saw that the behaviour of differential equations could be studied in terms of first return maps for points on the Poincaré section to itself. In this way the maps are related to differential equation systems.
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4. Monopoly

Abstract
Traditional microeconomic theory deals with two basic market types: Perfect competition and monopoly. In the case of perfect competition individual firms are assumed to be so small in comparison with the entire market that they cannot noticeably influence market price on their own; they just note the current price and react accordingly with respect to their supply. Only the supply of all the numerous firms together becomes a force on the market strong enough to determine the price in a balance with the demand of all the likewise numerous and small households.
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5. Duopoly and Oligopoly

Abstract
Economics recognizes two opposite market forms: competition and monopoly. In the competitive case the firms are very numerous and thus small in relation to the total size of the market. In consequence they consider market price as being approximately given independently of any action they can take on their own with regard to their supply.
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6. Business Cycles: Continuous Time

Abstract
The invention by Paul Samuelson in 1939 of the business cycle machine, combining the multiplier and the accelerator, certainly was a major event. That two such simple forces as consumers spending a given fraction of their incomes on consumption and producers keeping a fixed ratio of capital stock to output (=real income) combined to produce cyclical change was simple, surprising and convincing at the same time. This model if any qualifies for the attribute of scientific elegance. In passing it should be stressed that the Keynesian macroeconomic outlook was an essential background.
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7. Business Cycles: Continuous Space

Abstract
In economics there has been very scant modelling by means of partial differential equations. In business cycle modelling they, however, arise in a natural way if we put the problem studied in a spatial setting by introducing interregional trade, and considering a dynamical process that evolves both in continuous space and time. This will be done in the simplest and most obvious way, i.e., by a linear import-export multiplier, as is in line with the multiplier for local expenditures already present and with the general Keynesian macroeconomic outlook.
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8. Business Cycles: Discrete Time

Abstract
Although we for reasons given prefer to work with continuous time models it must be admitted that there are certain advantages in displaying the details of chaos for discrete time models. This is so because, before the tools of analysis, such as symbolic dynamics, can be applied to the continuous models we need to construct the return map on the Poincaré section for the orbit investigated. This, however, means that we first have to integrate the system over a complete cycle. The details of such an integration can easily become just much too complex.
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Backmatter

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