![Published by
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Library of Congress Cataloging-in-Publication Data
Names: Zhang, Wei-Bin, 1961- author.
Title: Chaos, complexity, and nonlinear economic theory / Wei-Bin Zhang,
Ritsumeikan Asia Pacific University, Japan.
Description: New Jersey : World Scientific, [2023] | Series: Series on
advances in mathematics for applied sciences, 1793-0901 ; vol. 92 |
Includes bibliographical references and index.
Identifiers: LCCN 2022048783 | ISBN 9789811267413 (hardcover) |
ISBN 9789811267420 (ebook) | ISBN 9789811267437 (ebook other)
Subjects: LCSH: Economics, Mathematical. | Nonlinear theories. | Chaotic
behavior in systems.
Classification: LCC HB135 .Z535 2023 | DDC 330.01/51--dc23/eng/20221021
LC record available at https://lccn.loc.gov/2022048783
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![List of Figures
Fig. 2.1 The bifurcation diagram for 15 ≤ β ≤ 4.7
with α = 0.7. . . . . . . . . . . . . . . . . . . . . . . . . . 25
Fig. 2.2 Specified relations between expected price and
supply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Fig. 2.3 Bifurcation diagram for μ = 0.5, a ∈ [−1.25, 1.25]. . . . . 26
Fig. 2.4 Bifurcation diagram for μ = 3, a ∈ [−1.25, 1.25]. . . . . . 27
Fig. 2.5 Bifurcation diagram for μ = 3.5, a ∈ [−1.25, 1.25]. . . . . 27
Fig. 2.6 Bifurcation diagram for μ = 4, a ∈ [−1.25, 1.25]. . . . . . 28
Fig. 2.7 Bifurcation diagram for μ = 4.5, a ∈ [−1.25, 1.25]. . . . . 28
Fig. 2.8 The dynamics with β = 0.2. . . . . . . . . . . . . . . . . 30
Fig. 2.9 The bifurcation diagram with β as bifurcation
parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Fig. 2.10 Bifurcation diagram of unemployment over the
inflow rate. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Fig. 2.11 An attractor. . . . . . . . . . . . . . . . . . . . . . . . . . 33
Fig. 2.12 A plot of x(t) for r = 28 for initial
condition (6, 6, 6). . . . . . . . . . . . . . . . . . . . . . . 35
Fig. 2.13 Plot of x(t) for r = 28. . . . . . . . . . . . . . . . . . . . 36
Fig. 2.14 The dynamics of the Lorenz equations. . . . . . . . . . . 37
Fig. 2.15 Projections of a trajectory of the Lorenz equations. . . . 37
Fig. 2.16 Bifurcation diagram with θ1 and the maximum Lyapunov.
(a) the firm 1, 2 of quantities trajectory in the market A,
(b) the firm 1, 2 of quantities trajectory in the market
B, (c) the cost trajectory of firm 1, 2, and (d) the profit
trajectory of firm 1, 2. . . . . . . . . . . . . . . . . . . . . 39
xv](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-20-320.jpg)
![List of Figures xvii
Fig. 3.9 Cycles generated in the Solow model by delays. . . . . . 75
Fig. 3.10 Bifurcation with the delay parameter. . . . . . . . . . . . 76
Fig. 3.11 The behavior before the system loses its stability
at τ = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Fig. 3.12 Persistent business cycles at τ = 5.65. . . . . . . . . . . . 77
Fig. 3.13 The motion of the economic system. . . . . . . . . . . . . 82
Fig. 3.14 Oscillations in the elasticity of substitution between two
varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Fig. 3.15 Oscillations in the fixed labor cost of the middle
goods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Fig. 4.1 The 4-period orbit for a = 4.9. . . . . . . . . . . . . . . . 88
Fig. 4.2 The existence of chaos for a = 5.75 with x0 = 0.4. . . . . 89
Fig. 4.3 The dynamics with different initial conditions, a = 5.75.
(a) x0 = 0.400. (b) x0 = 405. . . . . . . . . . . . . . . . . 89
Fig. 4.4 Small differences at the beginning signify much. . . . . . 90
Fig. 4.5 The map of bifurcations for a ∈ [2, 5.75]. . . . . . . . . . 90
Fig. 4.6 Fixed points for ρ <. (a) Two fixed points: z∗
1 ≤ −1/ρ <
z∗
2 . (b) Two fixed points: −1
ρ < z∗
1 < z∗
2. (c) Unique fixed
point: z∗
> −1/ρ (d) No existence of fixed point. . . . . . 93
Fig. 4.7 Stable business cycles at T = 2. . . . . . . . . . . . . . . 95
Fig. 4.8 Period-2 business oscillations at T = 2.03. . . . . . . . . 96
Fig. 4.9 Economic chaos at T = 2.06. . . . . . . . . . . . . . . . . 96
Fig. 4.10 Behavior with differential initial states and bifurcation
diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Fig. 4.11 Business cycles. . . . . . . . . . . . . . . . . . . . . . . . 98
Fig. 4.12 Bifurcation diagram with the bifurcation
parameter μ. . . . . . . . . . . . . . . . . . . . . . . . . . 98
Fig. 4.13 Existence of cycles without delay. . . . . . . . . . . . . . 100
Fig. 4.14 Existence of cycles with delay and without
diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Fig. 4.15 Existence of cycles with delay and diffusion. (a) D2 = 0.1;
populations of two species are oscillating. (b) D2 = 2;
Large diffusion rate of predator species population and
the extinction of predator species. . . . . . . . . . . . . . 101
Fig. 4.16 Population and magnitudes of COEs. . . . . . . . . . . . 102
Fig. 4.17 Motion of the economy without perturbations. . . . . . . 106
Fig. 4.18 The discrimination rate against women oscillates
periodically. . . . . . . . . . . . . . . . . . . . . . . . . . 107](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-22-320.jpg)
![Complexity Theory in Economics 5
geometry allows a relatively easy mental image to apprehend the essence of
a problem. For nonlinear problems, there is usually no simple and universal
geometry. Investigation was case by case.
Samuelson (1970) described the attitude of neoclassical economists as
“naturally tended to think of models in which things settle down to a
unique position independently of initial conditions. Technically speaking,
we theories hoped not to introduce hysteresis phenomena into our model, as
the Bible does when it says. ‘We pass this way only once’ and, in so saying,
takes the subject out of the realm of science and into the realm of genuine
history.” Nonlinear science treats “hysteresis phenomena” as a normal and
regular part of organic systems during its processes. Chaos is useful and
beneficial in some situations, like gaming machines and stock markets,
and is harmful and should be avoided like natural disasters and wars.
This attitude is explained by Mirowski (1990): “Neoclassical economics
was blocked from following physics into the realm of a serious formal
dynamics, including the formal structure of Hamiltonians, and instead into
the spurious pseudo-dynamics of ceteris paribus conditions. This inability
to emulate the core of the ideal of deterministic explanation tarnished
the entire program of imitating physics. . . . [T]he absence of a legitimate
dynamics also compromised the ideal of a scientific empiricism. . . . What
could it mean to attempt to fit neoclassical equilibrium displayed no
necessary stability from one moment to the next? Instead, most prominent
first- and second-generation neoclassicals were hostile to attempts to
import techniques such as least-squares estimation into economics; and
the earliest efforts in this area were pioneered by individuals skeptical of
neoclassical theory. . . . Such disputes over the meaning of scientific activity
also compromised claims of neoclassical theory to have attained ‘scientific
status.”
1.2 Chaos Theory or Complexity Theory as Revolution
in Sciences
There are great changes in sciences in association with nonlinear mathe-
matics and computer. Modern technology enables contemporary scientists
to see and simulate the complexity of the world unimaginable by humans
even a few decades ago. Modern nonlinear sciences create theories that
would have been treated as trivial or practically useless. Many theories
and assumptions in traditional sciences which had been accepted as being
general are considered limited and partial. Among these progresses in](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-30-320.jpg)
![26 Chaos, Complexity, and Nonlinear Economic Theory
Fig. 2.2. Specified relations between expected price and supply.
Fig. 2.3. Bifurcation diagram for μ = 0.5, a ∈ [−1.25, 1.25].
It can be shown that under these specifications, the market condition that
the demand equals supply is expressed by the following difference equation:
pe
(t + 1) = (1 − λ)pe
(t) +
aλ
b
−
λ arctan(μpe
(t))
b
≡ f(pe
(t)).
We will demonstrate the dynamic behavior of the model by simulation. In
the remainder of this section, we fix λ = 0.3, b = 0.25 and consider a as
a bifurcation parameter for different values of μ. In the case of μ = 0.5,
Figure 2.3 depicts the bifurcation diagram for a ∈ [−1.25, 1.25]. We see
that there is a unique fixed point for the map f.](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-51-320.jpg)
![Business Cycles and Chaos with Price Dynamics 27
Fig. 2.4. Bifurcation diagram for μ = 3, a ∈ [−1.25, 1.25].
Fig. 2.5. Bifurcation diagram for μ = 3.5, a ∈ [−1.25, 1.25].
Let us rise the value of μ to 3 and depict the bifurcation diagram with
a as the bifurcation parameter as in Figure 2.4. For low values of a, there
is a unique fixed point. Around a = −0.9, a period-doubling bifurcation
occurs. The stable orbit remains until a reaches 0.9 and then a period-
halving occurs. Thereafter, the system settles down again to a unique stable
equilibrium. The diagram is symmetrical about the origin because of the
characteristic of the arctan function.
Figure 2.5 depicts the bifurcation diagram when μ = 3.5. The figure
shows a doubling bifurcation into a period-four orbit, which then turns
into a period-two orbit and finally a stable equilibrium.](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-52-320.jpg)
![28 Chaos, Complexity, and Nonlinear Economic Theory
pe*
1.5
1
0.5
−0.5
−0.5 0.5 1
−1
−1
−1.5
Fig. 2.6. Bifurcation diagram for μ = 4, a ∈ [−1.25, 1.25].
Fig. 2.7. Bifurcation diagram for μ = 4.5, a ∈ [−1.25, 1.25].
In Figures 2.6 and 2.7, we depict respectively the bifurcation diagrams
when μ = 4 and μ = 4.5. We see that within the period-four orbit, chaos
occurs.
2.3 An Inventory Model with Rational Expectations
This example discusses a disequilibrium inventory model. This example
draws on Hommes (1991: Chapter 28; see also Zhang, 2006b: Chapter 4).
Actual labor employed, L(t), is given by the short side of the labor market,
L(t) = min{Ld
(t), Ls
(t)}, where Ld
(t) and Ls
(t) are respectively demand
and supply of labor. Assume Ls
(t) = c, where c is constant. Let yd
(t)](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-53-320.jpg)
![Comparative Mortality of Great Capitals.
Our recent alarm at the appearance and progress of the cholera in
London may have drawn the attention of many who had before
been accustomed to pass them by with indifference, to those
columns in the papers in which the reports of the Registrar-General
on the state of the public health are from time to time recorded.
But we are perhaps hardly yet sufficiently awake to the importance
and interest of the statistics there contained, any more than to the
value of the short and, at first sight, rather unintelligible tables
which embody, day after day, the meteorological phenomenon
collected in London from so many different points on our own coast
and those of adjacent countries. These last statistics have an
interest which does not yet belong to those which relate to the
public health, in that they embrace reports from so many distinct
places which can be compared together. We, of course, only publish
our own statistics of health, disease, births, and deaths; and we
have not yet seen our way to the information that might be
gathered by a comparison of our own condition in these respects
with that of others under similar circumstances. The interest and
value of such a comparison is obvious enough; and some of the
results which might be hoped from it, if it were systematically and
scientifically made, may be guessed at by the perusal of a thin
volume of less than two hundred pages, lately published in Paris by
M. Vacher, [Footnote 136] which at first sight may seem not to
promise very much except to professional readers, but from which
we shall take the liberty of drawing a few facts which certainly
seem worthy of the attention of the more general public.
[Footnote 136: Etude Médicale et Statistique sur la
Mortalité à Paris, à Londres, à Vienne et à New-York en
1865. D'aprés les Documens officiels, avec une Carte
Météorologique et Mortuaire. Par le docteur L. Vacher.
Paris: F. Savy, 1866.]](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-63-320.jpg)
![world in this respect. [Footnote 137] M. Vacher reckons the water
supply in ancient Rome as 1492 litres a day for each inhabitant; in
modern Rome it is 1040; in New York, 159; in Vienna. 134;
[Footnote 138] in Paris, according to the new system, 109; in
London, 132. But no city seems to have its houses so well
supplied as London; in Rome a great quantity of water is wasted,
being left to run away from the fountains, while the houses are not
conveniently provided with water. We suppose that our old friend
the house-cistern, against which we have heard so many
complaints lately, is not an essential feature in our system of house
supply.
[Footnote 137: M. Vacher attributes the salubrity of
Rome—for, considering its position, it enjoys remarkable
salubrity—to the abundance and good quality of its
water. Lancisi, who practiced there as a physician in the
last century, accounts for the longevity of its inhabitants
in the same way. At all events, remarks M. Vacher, il est
impossible de n'étre pas frappé de ce fait, que les
historiens ne mentionnent pas un seul example de peste
à Rome, et qu'au moyen age et dans les temps
modernes elle a constainment échappé aux atteintes de
la pests et du choléra, qui ont sévi à plusteurs reprises
en Italie. But Rome has certainly been visited by the
cholera more than once, and the rest of the statement is
surely contrary to history.]
[Footnote 138: This statement is, however, an
anticipation. The municipality of the Vienna has
undertaken some immense works in order to improve
the water supply, at a cost of 16,000,000 florins. The
works are not yet completed: but M. Vacher gives the
quantity of water for each inhabitant which they are
expected to furnish. Hitherto the city has been supplied,
it would seem, partly from the Danube, partly by wells.](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-67-320.jpg)
![The new supply will be drawn from three different
sources among the neighboring mountains.]
M. Vacher gives the following conclusions as to the sanitary effect
of good and abundant water. He tells us that inorganic substances
contained in water are comparatively innocuous to the health of
those who drink it; on the other hand, great injury is caused by the
presence of organic matter. The best water in Paris—that of the
springs on the north—contains nine times as much of calcareous
salts as the water of the Seine; but it is justly preferred for drinking
purposes. On the other hand, M. Vacher quotes the testimony of M.
Bouchut, a professor at the Ecole de Médecine, for the fact that he
noticed the frequency of epidemic diarrhoea during the summer
months in the Quartier de Sèvres and that it had been almost
stopped in cases where the doctors had ordered the water of the
Seine to be no longer used, and had substituted for it water from
the artesian well of Grenelle. He adds his own experience at the
Lycée Napoleon, which is supplied from the reservoir of the
Pantheon, which receives its water from the Seine and the
aqueduct d' Arcueil. He had known as many as fifteen students at
once ill of diarrhoea, and the disease was stopped by the
alcoholization of all the water. [Footnote 139]
[Footnote 139: P, 106. M. Vacher here cites the Indian
case quoted by Mr. Farre in is cholera report. The natives
in India drink boiled water as a preventative against
cholera; and it has been found that out of a great
number in the family of a single proprietor in Calcutta,
all of whom took this precaution, not a single person
had been attacked even in the worst times of the
prevalence of cholera. But Dr. Frank has disapproved at
least the universality of this fact.]
As regards cholera, the proof is even more striking than that lately
furnished in the case of London by the great and almost exclusive
ravages of that disease in the eastern districts. Mortality by cholera](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-68-320.jpg)
![geological formation as Montmartre, which had three times as
many deaths (in proportion) from cholera. In short, there is no way
left of accounting for its comparative exemption, except that which
we have already mentioned, the superior character of the water
consumed by its inhabitants. The argument certainly seems as
complete as it can possibly be, and we know that it has been
strongly confirmed by our own late experience. Let us hope that no
time may be lost in acting on the lesson which we have received.
We pass over some interesting statements on the meteorological
phenomena which were observed during the prevalence of the
cholera last year in Paris. [Footnote 140]
[Footnote 140: M. Vacher here tells a story of his
endeavor to make some ozonometrical observations in
the Paris hospitals, which were prohibited by the
Directeur de l'Assistance publique—an officer of whom
M. Vacher is continually complaining on the ground that
they would frighten the patients. He remarks that on
one occasion when travelling in the pontifical states,
some gendarmes found in his possession a psychrometer
and an aneroid barometer, and thought they were
weapons of destruction. He would have been arrested
but for M. Matteucci, then Director of Police. He
complains bitterly of the comparative want of
enlightenment in the administration of his own country.
But no hospital would have allowed his experiments.]
M. Vacher rather contradicts current opinion by some remarks he
has made as to the relation of cholera to other diseases. Sydenham
has remarked that when several epidemic diseases are rife during
the same season, one of them usually absorbs to itself, as it were,
the bulk of the mortality, diminishing the influence of the rest even
below the ordinary level. Thus in the year of the great plague in
London, just two centuries ago, the smallpox was fatal to only
thirty-eight persons, its average being about eleven hundred.](https://image.slidesharecdn.com/25675282-250509225741-e25406dc/85/Chaos-Complexity-And-Nonlinear-Economic-Theory-Weibin-Zhang-70-320.jpg)
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- 8. NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Series on Advances in Mathematics for Applied Sciences – Vol. 92 Wei-Bin Zhang Ritsumeikan Asia Pacific University, Japan CHAOS, COMPLEXITY, AND NONLINEAR ECONOMIC THEORY
- 9. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Zhang, Wei-Bin, 1961- author. Title: Chaos, complexity, and nonlinear economic theory / Wei-Bin Zhang, Ritsumeikan Asia Pacific University, Japan. Description: New Jersey : World Scientific, [2023] | Series: Series on advances in mathematics for applied sciences, 1793-0901 ; vol. 92 | Includes bibliographical references and index. Identifiers: LCCN 2022048783 | ISBN 9789811267413 (hardcover) | ISBN 9789811267420 (ebook) | ISBN 9789811267437 (ebook other) Subjects: LCSH: Economics, Mathematical. | Nonlinear theories. | Chaotic behavior in systems. Classification: LCC HB135 .Z535 2023 | DDC 330.01/51--dc23/eng/20221021 LC record available at https://lccn.loc.gov/2022048783 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13173#t=suppl Desk Editors: Logeshwaran Arumugam/Lai Fun Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore
- 10. Preface I first met with modern economics in 1984 when I was sent by China, as one of the second group of made-in-China undergraduates since 1949, to Japan in 1983 as a graduate student in civil engineering at Kyoto University. The library on campus was a wonderful world filled with freely available classical books for the poor student from the poverty-stricken country (when a mainland Chinese worker earned 7 USD per month in Beijing and Shanghai with the exchange rate in the early 1980s). The library was a pleasant and comfortable place for me to “kill” leisure time with philosophy, histories of sciences and mathematics, mathematics, mathematical biology, nonlinear sciences (typically, Prigogin’s and Haken’s works), as well as economics. I happened to read the works of some great modern economists like Samuelson and Arrow, luckily being introduced to mathematical economics with that the rigor, simplicity, creativity, beauty, passionate faith, and flexibility of mathematical economics of that generation. I passionately digested many modern and classical works in economic theory and had soon mathematically organized these theories. I was naturally attracted to classical economists, such as Adam Smith, Mill, and Marshall, because they explained economic principles with socioeconomic phenomena illustrative for me. The phenomena in their works were somehow like what took place in China in the 1970s and 1980s. It had taken me only few years to recognize the necessity and possibility of generalizing economic theory. Samuelson’s milestone achievements are reflected in two directions: one is summarized in his celebrated Foundation and the other is in his most celebrated textbook and numerical articles which mathematically formulate or generalize traditional economic ideas and v
- 11. vi Chaos, Complexity, and Nonlinear Economic Theory theories. Before Samuelson (more accurately his generations), mathematics was not yet applied so extensively and so systematically to economics. The Foundation basically contains what could be done mathematically with linear sciences by the time of its publication. I — equipped with modern nonlinear science and mathematical biology — soon extended Samuelson’s Foundation to nonlinear world with my first book Synergetic Economics (Zhang, 1991). The book was published in Professor Haken’s influential series Synergetics. My second goal, inspirited by Samuelson and some other economists, was to integrate all the main economic ideas from Adam Smith to contemporary Nobel Prizes in a single set of equations. The mission seemingly impossible was — at least as a somehow primary but integrative attempt — achieved with my recent The General Economic Theory (Zhang, 2020a) and other over one hundred articles published in journals. The construction of the general economic theory took me long time because I had needed even some years to digest and re-formulate each subfield of modern economics. Samuelson accounted his choice of economics at exactly the right time: “To a person of analytical ability, perceptive enough to realize that mathematical equipment was a powerful sword in economics, the world of economics was his or her oyster in 1935. The terrain was strewn with beautiful theorems begging to be picked up and arranged in unified order.” After I published my first two books, Synergetic Economics and my dissertation Economic Growth Theory (Zhang, 1990) simultaneously in 1989 (as research reports), I started to construct the general economic theory. In the Foreword to the Japanese translation of Synergetic Economics (Zhang, 1994), I outlined my aspiration to build the general economic theory in the following way: “Indeed, it is only after laborious work in many fields of theoretical economics that I began to be conscious of the fact that it is time to build a logically compact theory which includes the main economic ideas of Smith, Malthus, von Thunen, Ricardo, Marx, Mills, Walras, Marshall, Schumpeter, and Keynes. It should also include, as special cases, the well-established mathematical models, such as the Arrow–Debreu general equilibrium model, the Tobin model, the Solow–Swan–Uzawa growth model, the Oniki–Uzawa trade model, the Kaldor–Pasinetti two-class model, the Ricardian Models by Morishima, Samuelson and Pasinetti, the Keynesian theory, and Alonso location model, to explain certain economic phenomena which cannot be explained by the
- 12. Preface vii traditional works. I have concentrated on this single task, since . . . the spring of 1989.” This book is a continuation of my endeavor in complexity theory in economics. It is a continuation and “updating” of three of my previous books: Synergetic Economics. Heidelberg: Springer-Verlag, 1991. Differential Equations, Bifurcations, and Chaos in Economics. Singapore: World Scientific, 2005. Discrete Dynamical Systems, Bifurcations and Chaos in Economics. Else- vier: Amsterdam, 2006. Synergetic Economics was, perhaps, the first comprehensive book on apply- ing modern nonlinear theory and ideas from natural sciences to economics. It is a further development of Samuelson’s Foundation. Since 1990, there are numerical publications on nonlinear economics. My other two books, as the titles suggest, introduce new results in the field from 1990 till the early 2000s. They provide systematical introductions to nonlinear differential and difference equations theories which are fundamental to complexity theory in economics. This book gives a range of examples in recent advances in nonlinear economics. Most of these models are constructed by introducing some nonlinear elements to the traditional economic models which are mentioned in microeconomic or macroeconomic courses in undergraduate or graduate levels. To study complexity theory requires some years on advanced mathematics to digest technical details. Moreover, economics is composed of a great range of models/theories — each with its assumptions, refined structures, and complicated techniques. This book introduces some of recent developments in nonlinear economics with minimum mathematics and a plenty of simulation results with illustrative plots. I am very grateful to Editor Lai Fun Kwong for effective co-operation. I would like to thank the two anonymous referees for the valuable comments and suggestions. I thank for my wife, Gao Xiao, for caring. Daily digital- connected chats always warm my heart since March 2020 after the COVID- 19 started. The pandemic has kept many family members far away from each other. It is the stability and order that conquer external chaos and comfort the heart. There are some quotes which I don’t provide sources as almost all of them are from https://www.brainyquote.com, with a few
- 13. viii Chaos, Complexity, and Nonlinear Economic Theory exceptions by googling. I completed this book at the Ritsumeikan Asia Pacific University. From February 2020, I have been missing the beautiful campus and its lively international life. I am grateful to Editor Logesh for the efficient editorial work. Wei-Bin Zhang Beppu Summer 2023
- 14. About the Author Wei-Bin Zhang, Ph.D. (Umeå, Sweden), is Pro- fessor at Ritsumeikan Asia Pacific University since 2000. He graduated in 1982 from Beijing University. He obtained his Master’s Degree and completed his Ph.D. at the Department of Civil Engineering, Kyoto University in 1987. He completed his dis- sertation on economic growth theory in Sweden, 1989. Since then, he researched at the Swedish Institute for Futures Studies in Stockholm for 10 years. His main research fields are complexity theory in economics (nonlinear economic dynamics, chaos theory, and synergetic economics), history of ancient Chinese thought, Confucian- ism, American civilization, and economic development and moderniza- tion of Chinese societies. He single-authored 360 academic articles (180 in peer-reviewed international journals, https://ideas.repec.org/e/pzh151. html) and 29 academic books in English by well-known international academic publishing houses (https://www.amazon.com/Wei-Bin-Zhang/e/ B09LLS593T?ref=sr ntt srch lnk 1&qid=1648007965&sr=1-1). He is in the editorial board of 12 peer-reviewed international journals. He had published a book of poetry in Chinese by a well-known publisher in the field. Professor Zhang is the editor of Encyclopedia of Mathematical Models in Economics (in two volumes) as a part of the unprecedented global effort and The Encyclopedias of Life Support Systems, organized by the UNESCO. ix
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- 16. Contents Preface v About the Author ix List of Figures xv 1. Complexity Theory in Economics 1 1.1 Traditional Economics with Newtonian World View . . . 1 1.2 Chaos Theory or Complexity Theory as Revolution in Sciences . . . . . . . . . . . . . . . . . . . . 5 1.3 Synergetic Economics and Complexity Theory in Economics . . . . . . . . . . . . . . . . . . . . . 8 1.4 The Structure of the Book . . . . . . . . . . . . . . . . . . 16 2. Business Cycles and Chaos with Price Dynamics 21 2.1 A Cobweb Model with Adaptive Production Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Chaos in a Simplified Demand and Supply Model . . . . . 25 2.3 An Inventory Model with Rational Expectations . . . . . 28 2.4 Unemployment, Inflation, and Chaos . . . . . . . . . . . . 31 2.5 Urban Chaos by the Lorenz Equations . . . . . . . . . . . 33 2.6 A Cournot Game with Bounded Rationality and Behavioral Chaos . . . . . . . . . . . . . . . . . . . . 38 2.7 Chaotic Inflation and Unemployment with Taylor Rules . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.8 A Financial Dynamic Model with Delay . . . . . . . . . . 43 2.9 Stock Markets with Heterogeneous Agents . . . . . . . . . 45 2.10 Rent and Speculations in Housing Market . . . . . . . . . 47 xi
- 17. xii Chaos, Complexity, and Nonlinear Economic Theory 2.11 A Generalized Neoclassical General Equilibrium Theory with Tobin’s Model and the Taylor Rule . . . . . 50 2.12 Optimal Behavior . . . . . . . . . . . . . . . . . . . . . . . 54 3. Complexity of Economics with Capital and Wealth 57 3.1 Poverty Traps in an Extended Solow Model . . . . . . . . 58 3.2 Business Cycles in an Extended IS-LM Model . . . . . . . 61 3.3 A Keynesian Dynamics with Capital and Debts . . . . . . 62 3.4 Credit Cycles in an Extended Diamond Model . . . . . . 66 3.5 Business Cycles in a Goodwin–Kalecki–Marx Model . . . 71 3.6 Oscillations in a Reformed Solow Model with Delays . . . 73 3.7 A Reformed Kaldor–Kalecki Growth with Expectation and Delay . . . . . . . . . . . . . . . . . . . . 75 3.8 Neoclassical General Equilibrium Theory Generalized with Monopolistic and Perfect Competition . . . . . . . . 77 3.9 Demand and Supply of Final Goods . . . . . . . . . . . . 81 4. Nonlinear Population Dynamics 85 4.1 Chaos in a Generalized Malthusian Growth Model . . . . 86 4.2 Economic Dynamics with Endogenous Fertility and Old Age Support . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 Period Doubling Cascades of Prey–Predator Model . . . . 94 4.4 Chaos in a Discrete Prey–Predator Model . . . . . . . . . 96 4.5 Spatial Pattern of Prey–Predator Model with Delay in Prey . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.6 Carryover Effects on Population . . . . . . . . . . . . . . 101 4.7 Generalized Neoclassical General Equilibrium Theory with Generalized Malthusian Theory . . . . . . . . . . . . 103 5. Technological Changes and Human Capital Accumulation 111 5.1 A Generalized Solow–Uzawa Model with Poverty and Education . . . . . . . . . . . . . . . . . . . . 113 5.2 Oscillations between the Solowian and Schumpeterian Economies . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 Cycles of Work Hours Explained by New Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
- 18. Contents xiii 5.4 Homoclinic Bifurcation in a Modified Romer Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 Chaos in the Uzawa–Lucas Model . . . . . . . . . . . . . . 129 5.6 A Two-Stage Cournot Game with R&D Spillover . . . . . 132 5.7 A Cobweb Model with Memory and Competing Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.8 International Trade and Product Cycles in New Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.9 Generalized Neoclassical General Equilibrium Theory with Uzawa Two-Sector, Uzawa–Lucas Two-Sector, and Onik–Uzawa Multi-Country Models . . . . . . . . . . 144 5.10 International Trade . . . . . . . . . . . . . . . . . . . . . . 151 6. Economic Complexity with Environment and Resources 157 6.1 An Economic Growth Model with Pollution-Induced Poverty Traps . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.2 Poverty Trap with Natural Resources in a Solow–Ramsey-Based Model . . . . . . . . . . . . . . 161 6.3 Growth and Resources in an Overlapping Generations Model . . . . . . . . . . . . . . . . . . . . . . 164 6.4 Business and Environmental Cycles in a Solow–Ramsey-Based Model . . . . . . . . . . . . . . 167 6.5 The Role of Environment in Dynamics of Cournot Game . . . . . . . . . . . . . . . . . . . . . . . 169 6.6 Environmental Dynamics and Development in a Multi-Regional Economy . . . . . . . . . . . . . . . . 171 7. Complexity of Economies and Evolution of Economics 181 Bibliography 187 Index 199
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- 20. List of Figures Fig. 2.1 The bifurcation diagram for 15 ≤ β ≤ 4.7 with α = 0.7. . . . . . . . . . . . . . . . . . . . . . . . . . 25 Fig. 2.2 Specified relations between expected price and supply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Fig. 2.3 Bifurcation diagram for μ = 0.5, a ∈ [−1.25, 1.25]. . . . . 26 Fig. 2.4 Bifurcation diagram for μ = 3, a ∈ [−1.25, 1.25]. . . . . . 27 Fig. 2.5 Bifurcation diagram for μ = 3.5, a ∈ [−1.25, 1.25]. . . . . 27 Fig. 2.6 Bifurcation diagram for μ = 4, a ∈ [−1.25, 1.25]. . . . . . 28 Fig. 2.7 Bifurcation diagram for μ = 4.5, a ∈ [−1.25, 1.25]. . . . . 28 Fig. 2.8 The dynamics with β = 0.2. . . . . . . . . . . . . . . . . 30 Fig. 2.9 The bifurcation diagram with β as bifurcation parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Fig. 2.10 Bifurcation diagram of unemployment over the inflow rate. . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Fig. 2.11 An attractor. . . . . . . . . . . . . . . . . . . . . . . . . . 33 Fig. 2.12 A plot of x(t) for r = 28 for initial condition (6, 6, 6). . . . . . . . . . . . . . . . . . . . . . . 35 Fig. 2.13 Plot of x(t) for r = 28. . . . . . . . . . . . . . . . . . . . 36 Fig. 2.14 The dynamics of the Lorenz equations. . . . . . . . . . . 37 Fig. 2.15 Projections of a trajectory of the Lorenz equations. . . . 37 Fig. 2.16 Bifurcation diagram with θ1 and the maximum Lyapunov. (a) the firm 1, 2 of quantities trajectory in the market A, (b) the firm 1, 2 of quantities trajectory in the market B, (c) the cost trajectory of firm 1, 2, and (d) the profit trajectory of firm 1, 2. . . . . . . . . . . . . . . . . . . . . 39 xv
- 21. xvi Chaos, Complexity, and Nonlinear Economic Theory Fig. 2.17 Bifurcation diagram with θ1 and the maximum Lyapunov. (a) the firm 1, 2 of quantities trajectory in the market A, (b) the firm 1, 2 of quantities trajectory in the market B, (c) the cost trajectory of firm 1, 2, and (d) the profit trajectory of firm 1, 2. . . . . . . . . . . . . . . . . . . . . 40 Fig. 2.18 Chaos with m as the bifurcation parameter. . . . . . . . 42 Fig. 2.19 Strange attractor in u and πe with m = 16. . . . . . . . . 42 Fig. 2.20 Controlling strange attractors with the Taylor rule. . . . 43 Fig. 2.21 Chaos without time delay. . . . . . . . . . . . . . . . . . 44 Fig. 2.22 The bifurcation diagram for 0.01 ≤ μ∗ ≤ 0.03. . . . . . . 45 Fig. 2.23 The bifurcation diagram with γ as the bifurcation parameter. (a) λ = 1, (b) λ = 1.5, and (c) λ = 2. . . . . . 47 Fig. 2.24 Motion of the system bifurcated from cycles to chaos. (a) stable 2-cycle, γ = 0.35, (b) a pair of closed curves, γ = 0.4, (c) chaotic attractor, γ = 0.45, (d) γ = 0.35, (e) γ = 0.35, and (f) γ = 0.35. . . . . . . . . . . . . . . . 48 Fig. 2.25 Bifurcation diagrams of P with χ as the bifurcation parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Fig. 2.26 The motion of the system with wealth and money. . . . . 55 Fig. 2.27 Oscillatory perturbations in the targeted inflation rate. . . . . . . . . . . . . . . . . . . . . . . . . 56 Fig. 2.28 Oscillatory perturbations in the propensity to hold money. . . . . . . . . . . . . . . . . . . . . . . . . . 56 Fig. 3.1 Poverty trap in the extended Solow model. . . . . . . . . 60 Fig. 3.2 Hopf bifurcation near β = 6.385. (a) β = 5.8; (b) β = 6.385; (c) β = 7. . . . . . . . . . . . . . . . . . . 63 Fig. 3.3 The business cycle in the key variables. . . . . . . . . . . 66 Fig. 3.4 Oscillations in the firm’s debt and output. . . . . . . . . 66 Fig. 3.5 Business cycles. (a) μB = 0.2 (2-cycle), with w0 = 0.9, (b) μB = 0.125 (G2,4), with w0 = 0.9, (c) μB = 0.1125 (G2,2), with w0 = 0.9, (d) μB = 0.085 (G1), with w0 = 0.9, (e) μB = 0.032 (3-cycle), with w0 = 0.97, (f) μB = 0.0285 (G3,6), with w0 = 0.97, (g) μB = 0.0275 (G3,3), with w0 = 0.97, and (h) μB = 0.0245 (G1), with w0 = 0.99. . . . . . . . . . . . . . . . . . . . . . . . 69 Fig. 3.6 A bifurcation scenario in the parameter μB. . . . . . . . 70 Fig. 3.7 Cyclical paths. . . . . . . . . . . . . . . . . . . . . . . . . 73 Fig. 3.8 Stable business cycles. . . . . . . . . . . . . . . . . . . . . 74
- 22. List of Figures xvii Fig. 3.9 Cycles generated in the Solow model by delays. . . . . . 75 Fig. 3.10 Bifurcation with the delay parameter. . . . . . . . . . . . 76 Fig. 3.11 The behavior before the system loses its stability at τ = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Fig. 3.12 Persistent business cycles at τ = 5.65. . . . . . . . . . . . 77 Fig. 3.13 The motion of the economic system. . . . . . . . . . . . . 82 Fig. 3.14 Oscillations in the elasticity of substitution between two varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Fig. 3.15 Oscillations in the fixed labor cost of the middle goods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Fig. 4.1 The 4-period orbit for a = 4.9. . . . . . . . . . . . . . . . 88 Fig. 4.2 The existence of chaos for a = 5.75 with x0 = 0.4. . . . . 89 Fig. 4.3 The dynamics with different initial conditions, a = 5.75. (a) x0 = 0.400. (b) x0 = 405. . . . . . . . . . . . . . . . . 89 Fig. 4.4 Small differences at the beginning signify much. . . . . . 90 Fig. 4.5 The map of bifurcations for a ∈ [2, 5.75]. . . . . . . . . . 90 Fig. 4.6 Fixed points for ρ <. (a) Two fixed points: z∗ 1 ≤ −1/ρ < z∗ 2 . (b) Two fixed points: −1 ρ < z∗ 1 < z∗ 2. (c) Unique fixed point: z∗ > −1/ρ (d) No existence of fixed point. . . . . . 93 Fig. 4.7 Stable business cycles at T = 2. . . . . . . . . . . . . . . 95 Fig. 4.8 Period-2 business oscillations at T = 2.03. . . . . . . . . 96 Fig. 4.9 Economic chaos at T = 2.06. . . . . . . . . . . . . . . . . 96 Fig. 4.10 Behavior with differential initial states and bifurcation diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Fig. 4.11 Business cycles. . . . . . . . . . . . . . . . . . . . . . . . 98 Fig. 4.12 Bifurcation diagram with the bifurcation parameter μ. . . . . . . . . . . . . . . . . . . . . . . . . . 98 Fig. 4.13 Existence of cycles without delay. . . . . . . . . . . . . . 100 Fig. 4.14 Existence of cycles with delay and without diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Fig. 4.15 Existence of cycles with delay and diffusion. (a) D2 = 0.1; populations of two species are oscillating. (b) D2 = 2; Large diffusion rate of predator species population and the extinction of predator species. . . . . . . . . . . . . . 101 Fig. 4.16 Population and magnitudes of COEs. . . . . . . . . . . . 102 Fig. 4.17 Motion of the economy without perturbations. . . . . . . 106 Fig. 4.18 The discrimination rate against women oscillates periodically. . . . . . . . . . . . . . . . . . . . . . . . . . 107
- 23. xviii Chaos, Complexity, and Nonlinear Economic Theory Fig. 4.19 Women’s human capital oscillates periodically. . . . . . . 108 Fig. 4.20 The propensity to have children oscillates periodically. . . . . . . . . . . . . . . . . . . . . . . . . . 109 Fig. 5.1 Path-dependent economic evolution. . . . . . . . . . . . . 115 Fig. 5.2 The path-dependent development as the education is discouraged. . . . . . . . . . . . . . . . . . . . . . . . . . 117 Fig. 5.3 An increase in the propensity to save. . . . . . . . . . . . 118 Fig. 5.4 Growth governed by the Solow mechanism with sA < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Fig. 5.5 The Solow–Schumpeter mechanisms with 1 < sA < θ − 1. . . . . . . . . . . . . . . . . . . . . . . . 122 Fig. 5.6 Cycles with research and without research. . . . . . . . . 126 Fig. 5.7 The double-pulse homoclinic orbit. . . . . . . . . . . . . 129 Fig. 5.8 The parametric hypersurface for bifurcations. . . . . . . 130 Fig. 5.9 Irregular time-dependent motion of the economy. . . . . 131 Fig. 5.10 Irregular growth rates with changes in γ. . . . . . . . . . 132 Fig. 5.11 The bifurcation diagram and the largest Lyapunov exponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Fig. 5.12 Changes in attractors with different values of α2. . . . . 136 Fig. 5.13 The bifurcation diagram with β as the bifurcation parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Fig. 5.14 Curves of period-doubling (PD) and Neimark–Sacker (NS). . . . . . . . . . . . . . . . . . . . . 140 Fig. 5.15 Cycles and fractal structure. . . . . . . . . . . . . . . . . 141 Fig. 5.16 Annual growth rates of output and TFP in North and South. (a) am = 105; (b) am = 95.25; (c) am = 95. . . . . 145 Fig. 5.17 Growth in trade and cycles in R&D expenditures with am = 95.25. . . . . . . . . . . . . . . . . . . . . . . . . . 146 Fig. 5.18 The motion of the global economy. . . . . . . . . . . . . 153 Fig. 5.19 Fluctuations in the total factor productivity of the HDE’s tradable sector. . . . . . . . . . . . . . . . . . . . . . . . 154 Fig. 5.20 Fluctuations in the HDE’s propensity to receive education. . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Fig. 5.21 Fluctuations in the LDE’s propensity to save. . . . . . . 155 Fig. 6.1 The S-shaped production function. . . . . . . . . . . . . 160
- 24. List of Figures xix Fig. 6.2 A few combinations of savings and environment policy. (a) Phase Diagram (k, p) with s = 0.8, u = 0.01. (b) Phase Diagram (k, p) with s = 0.8, u = 0.2. (c) Phase Diagram (k, p) with s = 0.9, u = 0.01. (d) Phase Diagram (k, p) with s = 0.9, u = 0.02. . . . . . . . . . . . . . . . . 160 Fig. 6.3 The Hopf bifurcation at s = ŝ. . . . . . . . . . . . . . . . 161 Fig. 6.4 A way for the system to collapse at s = 0.58. . . . . . . . 162 Fig. 6.5 An attracting limit cycle around P∗ 1 . . . . . . . . . . . . 163 Fig. 6.6 Two attracting limit cycles around P∗ 1 and P∗ 2 . . . . . . . 164 Fig. 6.7 Existence of a unique fixed point. . . . . . . . . . . . . . 166 Fig. 6.8 Illustrations of global indeterminacy. . . . . . . . . . . . 166 Fig. 6.9 Bifurcation diagram and indeterminacy. . . . . . . . . . . 167 Fig. 6.10 Existence of limit cycles with endogenous environment. . . . . . . . . . . . . . . . . . . . . . . . . . 170 Fig. 6.11 Bifurcation diagram for the environmental policy. . . . . 171 Fig. 6.12 The motion of the national and regional economies. . . . 178 Fig. 6.13 Periodic oscillations in the population. . . . . . . . . . . 178 Fig. 6.14 The impact of population on amenity being periodically perturbed. . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Fig. 6.15 Region 1’s tax rate on consumption is periodically changed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
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- 26. Chapter 1 Complexity Theory in Economics The Nobel Prize in Physics 2021 was awarded “for groundbreaking contri- butions to our understanding of complex systems” according to the prize committee. Half of the prize money was jointly awarded to Syukuro Manabe and Klaus Hasselmann “for the physical modeling of Earth’s climate, quantifying variability, and reliably predicting global warming” and the other half to Giorgio Parisi “for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales.” The prize reflects the importance of complexity theory for the development of sciences and understanding nature and humanities. Complexity theory, which is almost exchangeable with chaos theory, synergetics, bifurcation theory, or catastrophe theory in this study — in economics (with formal mathematical modeling and computer simulation) — has been developed as an academic field since a few decades ago as soon as formal modeling of chaos was conducted in natural sciences and mathematics. Diffusion of ideas, methods, and techniques in natural sciences and mathematics to economics is nowadays quick. Complexity theory is theoretically and practically valid for dealing with interactions between politics, economics, cultures, and environment. 1.1 Traditional Economics with Newtonian World View Newtonian world view had been the dominant vision of economic science since Adam Smith created his theory of economic equilibrium with perfect competition and division of labor. Both his great books, Wealth of Nations and Theory of Moral Sentiments, are constructed with Newtonian approach to complicated problems. The analytical approach is characterized by dividing a complex system into various individual parts and treating each 1
- 27. 2 Chaos, Complexity, and Nonlinear Economic Theory unit as a whole system. His approach explains why he did not integrate his moral theory and economic theory — each theory creates a beautiful long-term stable equilibrium point, just like a pendulum of the clock of his time. There are oscillations — but they are created owing to exogenous or temporary forces. The system always either self-destroys (unstable) or finally achieves at a stationary state (moving smoothly at a constant speed) or static equilibrium point. In Newtonian vision, time-dependent phenomena, such as aperiodic motion, chaos, and bifurcation, — well observed — are considered negligible or exogenous. Economic systems are characterized by complex systems with close interdependence of multiple forces. Traditional science fails to systemat- ically explain well-observed economic phenomena. Faith is necessary for constructiveness and satisfaction. Over centuries, scientists shared a set of intuitive beliefs about changes. One of these beliefs is that simple systems would behave in simple ways. A pendulum follows a deterministic law, and its motion is stable and predictable in the long term. It also implies that well-observed complex behavior should be owing to complex causes. Visibly irregular behavior in stock markets must be governed by a multitude of independent components under the influences of random external forces. Human society is a collection of heterogeneous people. Man is collec- tively conformal with the group identity for co-survival. Personal behavior appears random and unpredictable. A social system made of many parts may behave with some regular pattern or predictable phenomena. In physics, an example is a gas in a container with the gas molecules as the parts. In a macro world, Newton’s law of motion can predict the regular motion of planetary orbits under certain conditions. Predictability in Newton physics is well applied to economics, especially by economists who had faith in nothing but free market, such as Hayek. His influential books are mostly constructed with the vision of traditional physics. In traditional scientific theory, if the present state of the system is known, it is possible to foretell how it will move in the future and to describe how it has moved in the past. When applying the determinism and irreversibility to human history, it implies that if we know the dynamic mechanisms of societies and sufficient information at any single point in time, it is possible to know its states one hundred years ago and one hundred years late. This vision of physical world is often referred to as Newtonian. It came from classical mechanics. Newton made a revolutionary contribution to the earlier development of classical mechanics. There are many mathematical methods invented by, for instance, Leibniz, Lagrange,
- 28. Complexity Theory in Economics 3 and Euler in the 17th century, to describe the motion of bodies of a system under the influences of forces. In the 18th and 19th centuries, more advances were made, leading to new fields, such as Lagrangian mechanics, Hamiltonian mechanics, and analytical mechanics. Classical mechanics are very successful when describing the motion of large objects that are not extremely massive. It is not effective when objects move at speeds approaching the speed of light. When objects are too small, quantum mechanics, a subfield of mechanics, is needed. Special relativity theory is necessary to study objects with speeds approaching the speed of light. General relativity theory is applied to study extremely massive objects. Classical economics since Adam Smith until modern times has been influenced strongly by natural sciences. Scientific thought and experiments have had a strong influence on economic thought. Toffler (1984) reasoned about the efficiency of modern economics: “One of the most highly developed skills in contemporary Western civilization is dissection: the split-up of problems into their smallest possible components. We are good at it. So good, we often forget to put the pieces together again.” The Indian parable of the blind men and an elephant, dated at least c. 500 BC, illustrates the point as well. The story tells that a group of blind men, who have never had any idea about elephant before, come across an elephant. They describe the elephant through touching, imaging, and thinking. Each blind man touches a different part of the elephant, and no two or more men touch the same part. Each blind man describes the elephant differently from the others. One described the elephant as being like a thick snake as his hand landed on the trunk. Another claimed the elephant as a kind of fan as he touched its ear. Others portrayed the elephant like a tree trunk, a rope, or a wall. Each blind man believes in himself and blames all the others as dishonest, cheating, and narrow-minded. They finally began to fight against each other. The globally well-known story has various versions and diffuses for different reasons. In some versions, the elephant is perceived as God. In Japan, for instance, it informs that common people tend to fail to appreciate properly a great man or his great work. The parable hints that humans tend to get addicted to the thought based on their limited and subjective experience and to ignore and even deny others’ experiences and ideas. Modern world is characterized by a refined division of labor and com- plicated social, economic, cultural, and political structures. Professionalism is conducted for the sake of professionalism and is historically applicable mainly to soldiers. But the attitude and spirit are applicable to almost
- 29. 4 Chaos, Complexity, and Nonlinear Economic Theory every profession in post-industrial or knowledge-based or information-based societies. Max Weber characterized modern societies as specialists without spirit and sensualists without heart. Modern people cannot find identity, consistency, trustfulness, and often not even social justice in social and economic life. For instance, during the current pandemic, different people see the tragedy with varied perspectives. A medical doctor may only suggest some probabilities of different consequences. One political party may neglect possible health consequences for people with low income and low social status but insist on economic benefits (of the rich) in the name of national benefits. The other party in rival may be concerned only with the importance of medical care for the poor. An economist believing in Adam Smith may suggest freedom and no government intervention. Another one believing in socialism may suggest government control. The list can be continued. No one knows what can occur with certainty, but many have great self-confidence in selling their own opinions. Economics evolves as analytical and illustrative tools are advanced. Classical economics is like painting, neoclassical economics is like photo, and modern economics is like movie. Adam Smith’s Wealth of Nations is full of sentences without mathematics and illustrations based on mathematical principles. His work makes modern reader feel inaccurate. His works are intuitively rich, like a poem, but have less law-like and empirically testable theory. Neoclassical theory is mostly like photo. Neoclassical economics was not widely tested with extensive data. Contemporary economics, treating the previous economics as special cases, are like movies owing to modern mathematics and computer. The reductionist approach to economics had been dominant in analytical economics. It fosters the detailed study of each school of economics. However, over the last 40 years or so, an alternative approach has emerged for the study of complex economic systems in association with recent advances in nonlinear science, mathematics, and computer. Keywords such as self-organization, complexity, nonlinear interdependence, emergence of new structures, path dependence, the whole being greater than the sum of parts, and chaos characterize this approach. Linearity means that the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now. More precisely, this is referred to in a differential or incremental sense: for a linear growth economy, the increase in GNP is proportional to the value which the economy is producing, with the growth rate exactly independent of how much the economy has already produced. Linear models enjoy an identical, simple geometry. The simplicity of this
- 30. Complexity Theory in Economics 5 geometry allows a relatively easy mental image to apprehend the essence of a problem. For nonlinear problems, there is usually no simple and universal geometry. Investigation was case by case. Samuelson (1970) described the attitude of neoclassical economists as “naturally tended to think of models in which things settle down to a unique position independently of initial conditions. Technically speaking, we theories hoped not to introduce hysteresis phenomena into our model, as the Bible does when it says. ‘We pass this way only once’ and, in so saying, takes the subject out of the realm of science and into the realm of genuine history.” Nonlinear science treats “hysteresis phenomena” as a normal and regular part of organic systems during its processes. Chaos is useful and beneficial in some situations, like gaming machines and stock markets, and is harmful and should be avoided like natural disasters and wars. This attitude is explained by Mirowski (1990): “Neoclassical economics was blocked from following physics into the realm of a serious formal dynamics, including the formal structure of Hamiltonians, and instead into the spurious pseudo-dynamics of ceteris paribus conditions. This inability to emulate the core of the ideal of deterministic explanation tarnished the entire program of imitating physics. . . . [T]he absence of a legitimate dynamics also compromised the ideal of a scientific empiricism. . . . What could it mean to attempt to fit neoclassical equilibrium displayed no necessary stability from one moment to the next? Instead, most prominent first- and second-generation neoclassicals were hostile to attempts to import techniques such as least-squares estimation into economics; and the earliest efforts in this area were pioneered by individuals skeptical of neoclassical theory. . . . Such disputes over the meaning of scientific activity also compromised claims of neoclassical theory to have attained ‘scientific status.” 1.2 Chaos Theory or Complexity Theory as Revolution in Sciences There are great changes in sciences in association with nonlinear mathe- matics and computer. Modern technology enables contemporary scientists to see and simulate the complexity of the world unimaginable by humans even a few decades ago. Modern nonlinear sciences create theories that would have been treated as trivial or practically useless. Many theories and assumptions in traditional sciences which had been accepted as being general are considered limited and partial. Among these progresses in
- 31. 6 Chaos, Complexity, and Nonlinear Economic Theory sciences, complexity theory, as far as economics is concerned, is the most important change. It is the first time in human history that economists can empirically test and simulate complex economic systems composed of interactions of multiple agents. Chaos theory makes scientific communities to re-examine classical sci- ences. It goes beyond the principles of uncertainty and relativity discovered by Werner Heisenberg and Albert Einstein. It “discovers” another side of nature and society which is characterized by instability, spontaneity, and unpredictability which have been treated as trivial in Newtonian sciences. Chaos theory demonstrates that simple systems give rise to complex and unpredictable behavior. A nonlinear system described by a single variable with some simple nonlinear interactions with fixed time-environment environment can lead to chaos. This is beyond anyone’s imagination before computer was applicable, even though this is almost common knowledge in contemporary sciences. The law of complexity has been identified in vast areas of physics, chemistry, biology, economics, politics, and human as well as animal group behavior. It has supplied science with a new vision that theories and ideas are created in separate scientific disciplines like physics and economics that treat classical Newtonian scientific explanations often as special cases. A typical example is Adam Smith’s economics. His theory is still valid under certain circumstances, but not as a general, not to say a universal, valid doctrine even for capitalism. He failed to recognize self-destructiveness and chaos in capitalism systems without government intervention. Keynes, smartly claiming that we are all dead in the long term when being asked about the survival of capitalist systems, proposed that the government can save capitalism by making proper interventions. Under his advocates, capitalist governments have been expanded to such a degree that the so-called capitalist economies are making far more intervention than some former socialist economies. Centralized planning economies, as properly advised by Hayek, was correct in pointing out unsustainability, inefficiency, and chaos of planned economies. Chaos on modern computer was first identified when American mete- orologist Edward Lorenz was playing weather games. He discovered that simple systems of just three variables can give rise to indeterministic phe- nomena. Seemingly random or negligible exogenous changes, which would not affect the trend significantly as Newtonian science tells, are rapidly multiplied, soon resulting in totally chaotic behavioral patterns. This is the pioneering discovery in chaos theory in association with the development of computer. Without computer, scientists had little chance to establish chaos
- 32. Complexity Theory in Economics 7 theory as a mainstream because even a simple question involves a huge amount of calculation. Chaos theory was serendipitously discovered when fractal theory was developed. Fractal geometry describes and measures the nonlinear forms of nature. It is important for the development of chaos theory. Together with relativity and quantum mechanics, chaos and fractals are claimed as the four greatest discoveries of the 20th century. Order versus chaos, deterministic versus stochastic, linear versus nonlinear, stable versus unstable, and predicable versus unpredictable are no more separate but the feature of complex systems. Mankind is now simultaneously experiencing profound ruptures in many fields: academics, human relations, national relations, family structures, gender relations, inequality in income and wealth, global environment, pandemics, wars, and the like. Chaos is a proper word to describe the epoch in which we are living. Chaos is around us all time, but chaos which is causing so many structural and unpredictable — often dramatically harmful — consequences in global scales and scopes is not recorded in human history. If one wants to interpret these with traditional thinking or ways of explanations, it is interpreted as caused by human evils and punishments from God (in the West) or the Heaven (in Chinese cultures). Nevertheless, modern sciences equipped with increasingly enhanced capac- ity of computing, available big data, theories, and ideas from complexity theory try to explore these phenomena as endogenous processes of interde- pendence between man and nature. Complex systems with new structures from instabilities can hardly be analytically and accurately described in traditional sciences both owing to lack of analytical methods and computing capacities. Complexity theory implies that laws or rules have no causal efficacy. They do not lead to ordered results or expected consequences. They merely maintain forced orders and consistent relationships in the system. This property can be illustrated by the game of chess. One cannot apply the rules to predict the history of a chess game. People cannot even predict, with certainty, the next move in a chess game. In this system, it is not only the rules of the game. It is also composed of the players and their situation-dependent decisions among a huge number of possible options at each movement. A theory, called synergetics, was developed by Hermann Haken (1977, 1983). This theory inspirited me to apply complexity theory to economics in the late 1980s. Synergetics explains the formation and self-organization of patterns and structures in open systems far from equilibrium. Self-organization occurs in a macroscopic system, consisting of multiple nonlinearly interdependent
- 33. 8 Chaos, Complexity, and Nonlinear Economic Theory subsystems. The system self-organizes, depending on external conditions. Haken developed the enslaving principle which implies that the dynamics of fast-relaxing (stable) modes is determined by the slow dynamics of, often, only a few “order-parameters” (unstable) modes. The macroscopic pattern is characterized by the unstable modes. Mathematically, self- organization means an enormous reduction of degrees of freedom. The reduction macroscopically leads to an increase in order. This macroscopic order is independent of the details of the microscopic interactions of the subsystems. Haken’s synergetics is applied to self-organization systems in physics, biology, chemistry, sociology, and economics. Haken (1981: Chapter 13) exemplified the principle as follows: “The statistical properties of laser light change qualitatively at the laser threshold. Below laser threshold, noise increases more and more while above threshold, it decreases again.. . . Below laser threshold, light consists of individual wave tracks which are emitted from the individual atoms independently of each other. Above laser threshold, a practically infinitely long wave track is produced. In order to make contact with other processes of self-organization let us interpret the processes in a lamp or in a laser by means of Bohr’s model of the atom. A lamp produces its light in such a way that the excited electrons of the atoms make their transitions from the outer orbit to the inner entirely independently of each other. On the other hand, the properties of laser light can be understood only if we assume that the transitions of the individual electrons occur in a correlated fashion.. . . Above laser threshold, the coherent field grows more and more, and it can save the degrees of freedom of the dipole movements and the inversion. Within synergetics, it has turned out that is a quite typical equation describing effects of self- organization... . This equation tells us that the amplitude of the dipoles, which is proportional to A, is instantaneously given by the field amplitude B(t) (and by the fluctuating force). This is probably the simplest example which has turned out to be of fundamental importance in synergetics and which is called slaving principle.” In recent researches, the phenomenon is identified in different fields. 1.3 Synergetic Economics and Complexity Theory in Economics Different from natural sciences and biology in which novel things are constantly discovered or created, there are rarely anything novel in complexity theory in economics as far as basic economic assumptions are
- 34. Complexity Theory in Economics 9 concerned; every new basic idea or basic mechanism has some sort of precedent or echo from the past. For instance, human motivations and personal behavioral patterns in modern times can be found in records of ancient Greek, Chinese, and other civilizations. There is nothing new under the sun that is applicable, roughly, to economics as far as personal behavior is concerned. Adam Smith argued: “Masters are always and everywhere in a sort of tacit, but constant and uniform combination, not to raise the wages of labor.” Nevertheless, economic phenomena are novel and varied greatly across nations. Like an infinite variety of music which is created by combinations of a few musical notes and an infinite number of paintings which result from synthesizing a few colors, economic phenomena come from a few motivations. These basic motivations and mechanisms had perhaps already been “discovered” by classical economists. Nevertheless, theoretical economics has advanced from grave to grave, as Paul Samuelson described. There are already many thousands of journals in economics and the growth of article publications is still in the stage of acceleration. But fast growth in the literature does not provide any “essential” novel insights into economic mechanisms (which have been verbally described by classical economists, conveniently assumed from Adam Smith). Advances in economics reveal new collective phenomena and economic structures which could not be dealt with by classical economists. Economic systems’ components, such as agriculture and industry, man and woman, family and society in large, and the like, interact in complicated relations in various environments. Complexity theory in economics refers to the study of complex economic systems. Nonlinear systems are also characterized by scale-dependent emergence: some properties are observable only when the system is large enough. Even if microscopic components are fully deterministic, macroeconomic behavioral patterns may be chaotic and unpredictable. In economics, many sectors, many households, and many countries interact in multiple interactions, culminating in a higher order of emergence of economic structures greater than the sum of all the parts. The theory studies these complex linkages at various scales. Every day we observe social and economic events which are driven by human crowd behavior. We observe crowd-like phenomena which emerge from interactions between people. Traffic jams emerge from commuters who compete for space on a road in rush hours. Market crashes in financial markets owing to interactions of financial traders. In a financial market, the spontaneous formation of a crowd of people who wish to sell can lead to a market crash characterized by the price falling dramatically in a short time. In recent
- 35. 10 Chaos, Complexity, and Nonlinear Economic Theory decades, environment has caused global concerns because global warming is caused by multiple actors across the earth. Environmental problems have always been in human history. But it is only in the recent decades that mankind can collectively destroy not only local environments but also global environments in unpredictable scales and scopes. Mankind is suffering in one way or another from disasters caused by environmental changes. Nonlinear science tells that traditional Newtonian theory cannot gen- erally predict planetary orbits. When applying this to economics, even if we understand the behavior of all individuals, we cannot be sure about the corresponding emergent phenomena from the people. Organized complexity, like planned economy in China in the late 1950s and early 1960s, is composed of non-random interactions between the parts. These organized relationships can create a structure which leads to phenomena unpredictable or unplanned. In China, many million people lost their lives owing to famines and misplanning in the chaos during the late 1950s and early 1960s. Aggregated or collective behavior manifests properties not carried by individuals. The phenomena “emerged” with an “invisible hand.” Such an unexpected phenomenon often occurs in complex systems such as politics, economics, and religions. What Werner Herzog said obviously includes economic world: “Civilization is like a thin layer of ice upon a deep ocean of chaos and darkness.” Size matters, as life tells us. If scale matters, a region with a small population may experience different dynamics from another region with a large population. For instance, Taiwan and mainland China, initially with almost identical population qualities in the late 1940s, have experienced quite different paths of modernization. Strong emergence implies that qualitative behaviors are not irreducible to the system’s constituent parts. The whole is not the sum of its parts. This further implies that the whole cannot be explained in terms of the parts. It would be misled to believe that it is possible to explain the dynamics of human affairs and physical world by finding out and applying simple fundamental laws. At each level of complexity, there emerge new properties and new structures which do not exist at other levels. The whole and individuals evolve together. Through dynamic interac- tions, new forms of society emerge which are not reducible to its past. History lost the capacity to recognize its paths. This property provides a country or society opportunities to make up its history according to its contemporary needs as there are multiple possibilities of the history. The be- havior of a dynamic system is not equal to the sum of the parts — the former can be larger or smaller than the latter, or both equal. Complexity theory
- 36. Complexity Theory in Economics 11 deals with social evolution from perspectives of adaption and development for survival in constantly changing environment. Evolution is characterized by nonlinearity and uncertainty, like human history. New political, social, economic, and sexual systems emerge through dynamic interactions of these subsystems within and outside. New structures emerge through self- organization, feedback, learning, creativity, with locking-ins, bifurcations, and chaos. Evolution is associated with exchanges of slaving and enslaving processes in the sense that the master variables become stabilized and predictable, while once non-key variables become activated and suddenly emerge as the leading modes. Simple cause-and-effect relationships assumed in traditional theories turn out as sources of confusions in empirical studies. The atomistic approach in theoretical economics, which had been the dominant mode of thinking, is becoming an obstacle to the evolution of economics. It is essential for economics to explain macro phenomena from micro levels. Persons make rational as well as irrational decisions. Individuals form households and families. There are multiple possibilities of connections that one person may have with other members of the society. Firms and organizations are in regions and cities. Interactions between regions and nations through different channels at various levels form evolutionary networking systems and new structures emerge through self-organization. Different from natural sciences, basic units or agents are self-organizing systems and cannot be described by purely random processes, like in statistical mechanics. Some nonlinear scientists apply concepts and tools from natural sciences but provide little insights into the complexity of human interactions because humans are treated as “atoms.” Another implication of complexity theory is that it enables sciences to integrate various subtheories into a unified theory. There is a necessity of economizing economics for eliminating redundancy between schools. Neurath (1983, 172–3) reasons for unifying the sciences: “. . . the special sciences themselves exhibit in various ways the need for such a unification. For example, the different psychological theories employ so many different terms and phrases that it becomes difficult to know whether they are dealing with the same subject or not. . . One of the most important aims of the Unity of Science movement is therefore concerned with the unification of scientific language. Distinct terms occur in different disciplines which nevertheless may have the same function and much fruitless controversy may arise in trying to find a distinction between them. . . . A large collection of terms has been gathered by the various sciences during the centuries, and it is necessary
- 37. 12 Chaos, Complexity, and Nonlinear Economic Theory to examine this collection from time to time, for terms should not be multiplied beyond necessity.” I have made efforts to integrate different schools of economics in a united framework. I proposed the general economic theory. Its basic purpose is not to build a set of equations that integrate basic macroeconomic patterns, which have been theoretically or empirically identified in traditional economic theories, with microeconomic foundations. The theory allows us to derive the behavior of identified macroscopic structures at least in principle. Economy is a complex adaptive system which is characterized by a dynamic network of interactions. It is adaptive in that individual agents and collective behavior mutate and self-organize in association with changes in micro events or collection of events. Agents interact, adapt, learn, and create in the system. We daily live in adaptive systems, such as cities, clubs, companies, regions, nations, traffic flows, political parties, wars, terrorist groups, social networks, climates, markets, governments, and industries. Systems such as animal swarms, social insect colonies, immune systems, and the cell and the developing embryo are all dynamically adaptive. There are some key terms, such as strategy, artifact, agent, population, system, type, variety, interaction pattern, physical space, conceptual space, selection, and success criteria, which are frequently used in modeling complex systems, identified by Axelrod and Cohen (2001). Strategy, for instance, refers to a conditional action pattern that indicates what to do under what circumstances. Artifact is considered as a material resource that has a definite location and can respond to the action of agents. According to, for instance, Cilliers (1998) and Turner and Baker (2020), a complex adaptive system can be characterized as follows (with US and China’s relations as examples by the author): (i) The number of elements is large, and these elements are dynamically interacting. Conventional models are not only impractical but fail to reveal the properties of the system. There are exchanges of information and physical variables that these relations evolve. The system is adaptive and resilient. It is simultaneously ordered and disordered. These interactions are not predetermined and complicated. Any element or subsystem in the system is affected and affects some other elements or subsystems. The system, which is composed of interdependency and great diversity, is self-organizing. The system operates sometimes orderly and sometimes chaotically. During the Deng period, the relations between USA and China were orderly. Even
- 38. Complexity Theory in Economics 13 after the Beijing government conducted the military action against people’s demand for transparency and more democratization, the relationship between the two sides self-organized to regular patterns after a short period of the event. As far as international relation with the USA is concerned, China’s government was a relatively simple and linear system at least by the early 2000s, even though domestic politics within the party has been always complicated. Its behavioral patterns in international relations were more predictable than today. (ii) The interactions are nonlinear. Small changes in inputs, physical inter- actions, or stimuli can cause large effects or very significant changes in outputs. The motion is not predetermined but path-dependent. Hence, the system is very sensitive to its temporary conditions. The same force might affect the system differently. For instance, China’s reaction to American policy changed dramatically over the last 40 years, although China was dominated by the same party. America could have hardly “predicted” China’s behavior, especially in the last one or two decades. The reason was that even China could not have a solid and principle-based policy towards America. The two sides entered “unstable zones” of interactions so that everything was path-dependent, even though the chaotic interactions have not led to mutually self-destructive consequences. No one knows what will happen with interactions even in near future, but the probability of nuclear wars does not “appear” to be high because such a war will bring benefits neither to USA nor to China but almost certainly to their common future competitors, such as Europe, after their political and economic influences are weakened. (iii) Interactions are not exclusively with immediate neighbors. In an open system like the relationship between USA and China, it involves al- most the entire world in one way or another through political, cultural, social, economic, and even psychological channels. In particular, as digital technologies and advanced transportation systems with low costs have closely connected the world, no country could be effectively closed. It is nowadays even difficult to define system boundaries. National relations themselves are switched rapidly. No principle or value system could provide an effective foundation for nations to work as a team. (iv) Any interaction can feed back onto itself directly or after several interactions with other parts of the system or the environment. It may take some time to feed back. In human societies, collective memory
- 39. 14 Chaos, Complexity, and Nonlinear Economic Theory can last over hundred years. One reaction can also recall memories and thus make the consequence even more unpredictable. In peasant and almost immobile societies, memories of human relations and actions could last some generations. Complex systems have history. Even if the system has memory, it cannot go back to its past. The system is irreducible in the sense that the process is irreversible, and the current cannot be reduced back to its original state. For instance, owing to his “frankness,” President Trump said many negative things that mainland Chinese people could never dream about America’s “genuine” attitude towards people and the nation who had only known a little about modern civilization. His impact has made China’s image and masses’ mindsets on USA irreversible. The general trust on and the idol role of USA by common Chinese people are perhaps lost forever owing to the influence of Trump’s government. It is not owing to the difference between the Trump government and the other American governments but the Trump style which simplified America’s approach to China that even common Chinese people could perceive some hidden motivations of the political world. (v) The overall behavior of the system is not predicted by the individual agents. There is no way to sum or aggregate the behavior of individuals to accurately predict the dynamics of the whole. America had the best experts in multiple fields about China in the world. Every year there have been numerous books, articles, and news on China. But none could have predicted the sudden rise of China in an English-speaking world. If that could be done, China might not be able to rise so rapidly because not only the white world but also Japan and overseas Chinese societies would join to present the occurrence. All parts of China were understood, but the performance of the whole was unpredictable. (vi) Complex systems operate under far from equilibrium conditions. For life in the long term, a truly equilibrium condition is the ending of life. Life needs constant flow of energy to operate the system. Living systems are adaptive and not necessarily always “progressive.” The evolution is not characterized by being progressive and moving towards “higher organisms.” China from 1840 to 1980 could be characterized by degeneration rather than progressiveness. There were few higher cultural products left during the period. America could hardly be called “progressive” as an organic system in the last two decades, even though it has made many great advances in technologies. The fruits of these technological changes might not
- 40. Complexity Theory in Economics 15 benefit American people but further enrich a limited number of rich people and talented immigrants born outside America. (vii) Agents in the system may be ignorant of the behavior of the entire system. They react only to the information or physical stimuli available to them locally. Democracy, freedom, and transparency play an important role, it is assumed by liberal thinkers, in maintaining a decent society. Thinkers, like Lao Zi or Albert Einstein, might consider differently. For instance, Lao Zi believed that common people cannot properly understand Dao and it would benefit the society if they were not informed “too much.” Einstein does seem to prefer some authorial organization to direct the society. The system includes mechanisms of emergent structures. Its internal dynamics affect its ability to change in a manner that might be quite different from its history or other systems. Appearance of new industries and new types of products is a daily part of modern economy. It is generally true that evolution includes internal mechanisms towards complexity but does not necessarily lead to sustainability and survival. As the system has become too complicated, self-destruction is a solution. No great civilization has enjoyed a long period of desirable states. Traveling ancient traces of great civilizations, one can only see ruins or things hidden in graves which record the glory. Japan and America have become far more complex than their states in the mid-1980s, but their sustainability (with regard to common people’s living standards and environment) is difficult to be true. Mankind might create such a complex system based on future technologies that man evolves into a kind of man that modern man would not see the future man as being human. The meaning of evolution is thus meaningless for contemporary people. All enriched economies with highly educated women experienced rapid population declination. In a self-organized system, human life itself becomes meaningless except for being changeable. In Chinese long-recorded history, complexity built by human society would lead to destruction for simplification and re-constructing the meaning of human life. China’s Malthus cycles were repeated over thousand years without few significant social progresses. Many countries have remained poor, even though many modern elements have been injected into the systems. Modern economic miracles of Japan, then overseas Chinese societies, and now mainland China have been achieved owing to their inner Confucian cultural stocks and “fortune” initial conditions of poverty and low education (which enabled them to obediently follow the superpower’s “master mode”).
- 41. 16 Chaos, Complexity, and Nonlinear Economic Theory 1.4 The Structure of the Book The rest of this book is organized into six chapters. Each chapter, except Chapter 7, focuses on the dynamics of some well-known models with a key variable and finally introduces a general model into which main economic mechanisms of the previous models in the chapter are or can be integrated. Hence, the general model shows a possibility of building more general models for exploring the complexity of economic systems. Generalization is challenging but not impossible because mathematical tools are available. This book is mainly concerned with what economic mechanisms and mathematical structures nonlinear phenomena can be identified and illustrated. We explain mathematical models not in detail because most of the basic economic mechanisms are already taught in basic courses in microeconomics and macroeconomic courses for undergraduates. Moreover, the book does not provide analytical results and mathematical proofs. Proofs are often tedious and can be found in the original sources. Chapter 2 is concerned with price dynamics which are “classical” mech- anisms for explaining business cycles. Section 2.1 deals with the nonlinear cobweb model with normal demand and supply, naı̈ve expectations, and adaptive production adjustment proposed by Onozaki et al. (2000). It shows how the simple economic mechanism manifests equilibrium, periodic solution, aperiodic oscillations, and chaos. Section 2.2 demonstrates chaos in a simple demand–supply model. This example draws on Hommes (1991). Supply involves a time lag. At low prices, supply increases slowly, partly because of start-up costs and fixed costs of production. Supply might also increase only slowly at high prices, say, because of capacity constraints. This suggests an S-supply curve. Section 2.3 discusses a disequilibrium inventory model by Hommes (1991). Actual labor employed, L(t), is given by the short side of the labor market. Section 2.4 introduces a worker flow model with a nonlinear outflow rate from unemployment by Neugart (2004). Section 2.5 shows that urban dynamics of an open city can be described by the Lorenz equations. It shows urban chaos in population, land rent, and the city’s GDP. Section 2.6 explains the dynamic multi- market Cournot model proposed by Zhao et al. (2019). It analyzes the behavior of oligopolistic producers, applying an incomplete information approach to generalize the traditional approach in which each producer applies the expectation to suppose that the opponents’ output keeps the same level as previous period’s and adopts an output strategy to maximize the expected profit. Section 2.7 demonstrates chaotic inflation
- 42. Complexity Theory in Economics 17 and unemployment in a model Taylor Rules proposed by Guevara and Escot (2021). Section 2.8 studies the relatively simple model for financial markets with delays examined by Ding et al. (2012). Section 2.9 studies the model by Agliari et al. (2018) who deal with a stock market entry model in which participation depends on an attractiveness measure related to market activity and the fundamental value of the market. Section 2.9 introduces a housing market model with rent control proposed by Schmitt and Westerhof (2022). In this approach, homebuyers are assumed to be boundedly rational and able to form expectations with learning. They can switch between extrapolative and regressive expectation rules under the influences of their past forecasting accuracy. Policymakers may use rent control to affect the rent level. Section 2.10 considers the generalized neoclassical general equilibrium theory with Tobin’s model and the Taylor rule. It shows how period exogenous inputs generate business cycles in economic growth. Chapter 3 studies the complexity of economies with endogenous cap- ital and wealth. Chaos appears under different mechanisms of capital accumulation. Section 3.1 applies an approach based on Ros (2000) to demonstrate a poverty trap in an extended Solow model. The model has two equilibria: a Solow-type and an unstable steady state at the level of the capital stock comparable with subsistence consumption. Section 3.2 studies the extended IS-LM model by Ma et al. (2017) which exhibits Hopf bifurcation. Section 3.3 shows the existence of chaos in the Keynesian dynamic model with capital accumulation and debts recently developed by Asada et al. (2019). Section 3.4 shows endogenous oscillations of debts in a model proposed by Matsuyama (2013). In Section 3.5, we deal with the disequilibrium macrodynamic model by Sasaki (2013). The model incorporates the three important economic mechanisms for approaching modern capitalist economies: ideas on interactions between the rate of employment and income by Goodwin, investment function by Kalecki, and the reserve-army and reserve-army-creation effects by Marx. Section 3.6 demonstrates oscillations in the reformed Solow model with two delays by Guerrini et al. (2019). Section 3.7 shows nonlinear phenomena in the reformed Kaldor–Kalecki growth model with capital stock’s anticipation and investment lag analyzed by Liu et al. (2015) to show that can lead to the emergence of business cycles. Section 3.8 represents a generalized neoclassical general equilibrium theory integrated with monopolistic and perfect competition. The system is subject to various exogenous shocks in technologies and preferences.
- 43. 18 Chaos, Complexity, and Nonlinear Economic Theory Population causes complex phenomena in economies. It interacts with wealth, gender relations, education, and other factors. Chapter 4 presents a few nonlinear population dynamics. Section 4.1 introduces a population dynamic model constructed by Haavelmo in continuous form. Its discrete form was examined by Stutzer (1980), by applying modern mathematics for one-dimensional mappings. It is an early model that shows economic chaos from a simple economic model. Section 4.2 represents the nonlinear OLG model with endogenous fertility and old age support proposed by Chakrabarti (1999). Section 4.3 presents period-doubling cascades in a traditional prey–predator modeling framework developed by Gupta et al. (2014). Section 4.4 studies the discrete prey–predator model with chaos by Chen and Chen (2012). Section 4.5 represents the existence of nonlinear phenomena in a spatial extension of the classical prey–predator model with time delay in prey by Rao (2018). Section 4.6 represents the ecological model by Ruiz-Herrera (2017) that explores the effects of carryover effects (COEs) — any process that in one season that affects, non-lethally, individuals in the following seasons — on population dynamics. Section 4.6 provides a model of generalized neoclassical general equilibrium theory with generalized Malthusian theory. Chapter 5 is concerned with models with endogenous changes and human capital accumulation, which exhibit nonlinear phenomena. Sec- tion 5.1 represents the model by Zhang (2006a) which explains path- dependent economic growth. The model synthesizes Solow’s physical capital accumulation, Arrow’s learning-by-doing, and Uzawa’s learning-by- education growth models within a comprehensive framework. It is well known that the neoclassical growth theory based on the Solow growth model focuses accumulation of physical capital as an engine of growth, while the neo-Schumpeterian growth theory stresses innovation. Section 5.2 shows oscillations between the Solowian and Schumpeterian economies in the model proposed by Matsuyama (1999). Declining hours of work during one’s lifetime is a well-observed phenomenon among developed economies. In recent years, there are many models proposed to explain the economic mechanisms of endogenous labor supply in association with economic development. Section 5.3 represents a new-growth-theory model for cycles of works by Iong and Irmen (2021). Section 5.4 shows homoclinic bifurcation in the modified Romer growth model proposed by Bella (2017). Section 5.5 shows the existence of chaos in the Uzawa–Lucas model identified by Bella et al. (2017). Section 5.6 studies the dynamical behavior of the model for a two-stage Cournot duopoly game with R&D spillover effect and
- 44. Complexity Theory in Economics 19 product differentiation proposed by Zhou et al. (2019). The original paper investigates the behavior of the model, by applying Jury criterion, central manifold theorem, norm form theory, as well as numerical simulation. Section 5.7 introduces the model by Agliari et al. (2018), which is based on the cobweb demand–supply framework with costly innovators and free imitators. Section 5.8 introduces the international trade model with product cycles and growth cycles by Iwaisako and Tanaka (2017). Section 5.9 provides the model of generalized neoclassical general equilibrium theory which integrates Uzawa two-sector, Uzawa–Lucas two-sector, and Onik– Uzawa multi-country models within a comprehensive framework. Chapter 6 studies economic complexity with environment and resources. Section 6.1 represents the economic growth model with endogenous envi- ronment by Liuzzi and Venturi (2021). It explains the existence of pollution- induced poverty traps. Section 6.2 studies the model by Antoci et al. (2011) who apply global analysis techniques to economic growth development with natural resources within the Ramsey growth modeling framework. Section 6.3 presents the model of economic growth and environmental resources within an overlapping generations framework proposed by Antoci (2016). Section 6.4 introduces the model by Bosi and Desmarchelier (2018), which is built by integrating the Levins model and the Solow–Ramsey model with pollution externality. Section 6.5 studies the Cournot game model built by Fanti (2015). The model deals with the impact of public environmental policies in a Cournot duopoly game with heterogeneous expectations and limited rationality. Section 6.6 introduces the model of environmental dynamics and development in a multi-regional economy. Chapter 7 concludes the study.
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- 46. Chapter 2 Business Cycles and Chaos with Price Dynamics What Andrew Carnegie advised for money management seems spiritually effective for faith in God or professionally useful for some discipline: “The Way to become rich is to put all your eggs in one basket and then watch that basket.” The question is to be fortunate enough to have a secured basket. If a nation puts the political power to a politically inexperienced man through the masses’ elections, the basket is secured but the nation might get globally humiliated or even be broken. With regard to the role of money in economic evolution, Robert Lucas — who received the Nobel Prize in Economics in 1995 “for having developed and applied the hypothesis of rational expectations, and thereby having transformed macroeconomic analysis and deepened our understanding of economic policy” — stated his Newtonian view: “The central predictions of the quantity theory are, in the long run, money growth should be neutral in its effects on the growth rate of production and should affect the inflation rate on a one-for-one basis.” He got the Nobel prize because of this Newtonian vision on economic evolution. Another Nobel Prize winner, Paul Krugman, educated in classical economics and being brought up in the golden American epoch, stated his knowledgeable confidence: “We know that advanced economies with stale governments that borrow in their own currency can be capable of running up very high levels of debts without crisis.” Anything is rationally possible in theoretical economics as far as the economist enjoys freedom of speech without social (especially international) responsibility. Galbraith argued: “Nothing so weakens a government as inflation.” Henry Ford knew what he observed: “It is well enough that people of the nation do not understand our banking and monetary system, for if they did, I believe there would be 21
- 47. 22 Chaos, Complexity, and Nonlinear Economic Theory a revolution before tomorrow morning.” Stock markets are a game between millions of blind or ignorant players (with varied faiths or confidences) and a few (relatively) well-informed and experienced players/controllers. Stock markets, like romantic affairs, are typically chaotic if they are playful. We have little idea yet about what robots can know and do in the future. Prices and value of money are determined by emotions, rational calculations, technologies, market structures, and so on. Money, once wrongly issued for solving temporary national problems or political gains, might cause the national economy to slowly but steadily, or suddenly, deviate from its long-term healthy development path through labyrinthine accumulated effects that resulted from interdependence among almost-all important variables in political and economic systems. Benjamin Franklin said: “In this world nothing can be said to be certain, except death and taxes.” Chaos theory demonstrates relations between chaos and order through nonlinear relations. This chapter is concerned with price dynamics which are “classical” mechanisms for explaining business cycles. Section 2.1 deals with nonlinear cobweb model with normal demand and supply, naı̈ve expectations, and adaptive production adjustment proposed by Onozaki et al. (2000). It shows how the simple economic mechanism manifests equilibrium, periodic solution, aperiodic oscillations, and chaos. Section 2.2 demonstrates chaos in a simple demand–supply model. This example draws on Hommes (1991). At low prices, supply increases slowly, partly because of start-up costs and fixed costs of production. Supply might also increase only slowly at high prices, say, because of capacity constraints. Section 2.3 discusses a disequilibrium inventory model by Hommes (1991). Actual labor employed, L(t), is given by the short side of the labor market. Section 2.4 introduces a worker flow model with a nonlinear outflow rate from unemployment by Neugart (2004). Section 2.5 shows that urban dynamics can be described by the Lorenz equations. It shows urban chaos in population, land rent, and the city’s GDP. Section 2.6 explains the dynamic multi-market Cournot model proposed by Zhao et al. (2019). It analyzes the behavior of oligopolistic producers, by applying an incomplete information approach to generalize the traditional approach in which each producer applies the expectation to suppose that the opponents’ output keeps the same level as previous period’s and adopts an output strategy to maximize the expected profit. Section 2.7 demonstrates chaotic inflation and unemployment in a model Taylor Rules proposed by Guevara and Escot (2021). Section 2.8 studies the relatively simple model for financial markets with delays examined by Ding et al. (2012). Section 2.9 studies
- 48. Business Cycles and Chaos with Price Dynamics 23 the model by Agliari et al. (2018) who deal with a stock market entry model in which participation depends on an attractiveness measure related to market activity and the fundamental value of the market. Section 2.9 introduces a housing market model with rent control proposed by Schmitt and Westerhof (2022). In this approach, homebuyers are assumed to be boundedly rational and able to form expectations with learning. They can switch between extrapolative and regressive expectation rules under the influences of their past forecasting accuracy. Policymakers may use rent control to affect the rent level. Section 2.10 considers the generalized neoclassical general equilibrium theory with Tobin’s model and the Taylor rule. It shows how period exogenous inputs generate business cycles in economic growth The Nobel Prize in Physics 2021 was awarded “for groundbreaking contributions to our understanding of complex systems” according to the prize committee. Half of the prize money was jointly awarded to Syukuro Manabe and Klaus Hasselmann “for the physical modeling of Earth’s climate, quantifying variability and reliably predicting global warning” and the other half to Giorgio Parisi “for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales.” The prize reflects the importance of complexity theory for the development of sciences and understanding nature and humanities. Complexity theory, which is almost exchangeable with chaos theory, synergetics, bifurcation theory, or catastrophe theory in this study — in economics (with formal mathematical modeling and computer simulation) — has been developed as an academic field since a few decades ago as soon as formal modeling of chaos was conducted in natural sciences and mathematics. Diffusion of ideas, methods, and techniques in natural sciences and mathematics to economics is nowadays quick. Complexity theory is theoretically and practically valid for dealing with interactions between politics, economics, cultures, and environment. 2.1 A Cobweb Model with Adaptive Production Adjustment Adam Smith explained the foundation of price determination: “People in the same trade seldom meet together, even for merriment and diversion, but the conversation ends in some contrivance to raise prices.” But his analytical conclusion on price dynamics is valid only in limited situations. A simple nonlinear cobweb model — even without periodic exogenous inputs — illustrates the invalidity of his thought. A nonlinear cobweb
- 49. 24 Chaos, Complexity, and Nonlinear Economic Theory model with normal demand and supply, naı̈ve expectations, and adaptive production adjustment was proposed by Onozaki et al. (2000; see also Zhang, 2006b, Section 5.2). Let us consider a market of a single commodity. In period t, a supplier decides his production x(t+1), which may not equal the profit maximum x̃(t + 1). The optimal level is calculated and used as a target of adjustment by the supplier. Suppose that the calculation is made under the quadratic cost function (bx ∧ 2)/2, b > 0 with the naı̈ve price expectation (which means that his price expectation for the next period is equal to the current price p(t)). The profit maximum level of output is given by x̃(t + 1) = p(t)/b. It is assumed that the producer will adjust his production according to the following hedging rule in the uncertain economy x(t + 1) = x(t) + α(x̃(t + 1) − x(t)), where α ∈ (1) is the speed of adjustment. Suppose that there are n identical suppliers in the market. The aggregate supply is thus given by X(t) = nx(t). Assume a monotonic demand function with constant price elasticity of 1/β(β > 0) : p(t) = c/(Y ∧ β(t)). Price clears the market in each period, i.e., X(t) = Y (t). It is straightforward to show that under the above specifications, the motion of the aggregate supply is given by X(t + 1) = (1 − α)X(t) + αcn bXβ(t) . Introduce a linear transformation z(t) ≡ (b/cn)1/(1+β) X(t). Then, the above equation is transformed into z(t + 1) = (1 − α)z(t) + α zβ(t) ≡ f(z(t), α, β), (α, β) ∈ (0, 1) × (0, ∞). For this map, it can be demonstrated that for sufficiently large β, the map f exhibits a horseshoe. By a horseshoe, it means here a compact invariant set on which some iterate of f is topologically conjugate to the one-sided full-shift on two symbols. The existence of a horseshoe is assured by that of a transverse homoclinic point. A map is said to exhibit topological chaos if it has a horseshoe or, alternatively, if the topological entropy of the map is positive. It should be remarked that a map restricted to horseshoes behaves in a complicated way; the existence of horseshoes itself does not assume complex dynamics in the long run; the system may eventually settle down to a periodic motion even if horseshoes are present. In the following theorem proved by Onozaki et al., an attractor is said to be strange if it
- 50. Business Cycles and Chaos with Price Dynamics 25 Fig. 2.1. The bifurcation diagram for 15 ≤ β ≤ 4.7 with α = 0.7. contains a dense orbit with a positive Liapunov exponent. Figure 2.1 depicts the bifurcation diagram of the map with regard to β (15 ≤ β ≤ 4.7) with α = 0.7. 2.2 Chaos in a Simplified Demand and Supply Model We now demonstrate chaos in a simple demand–supply model. This example draws on Hommes (1991: Section 5.1; see also Zhang, 2006b: Section 4.6). We consider that supply involves a time lag. It should be remarked that an analytical framework proposed by Bellomo et al. (2020) can be applied to model cases of multiple goods. At low prices, supply increases slowly, partly because of start-up costs and fixed costs of production. Supply might also increase only slowly at high prices, say, because of capacity constraints. This suggests an S-supply curve. The arctan function exhibits such an S-shape. Let us express supply qs (t) as a function of expected price pe (t) by such a function: qs (t) = arctan(μpe (t)). The origin is an inflection point. As shown in Figure 2.2, the parameter μ determines the steepness of the S-shape. The higher the value of μ, the steeper the curve. It is assumed that demand is a linear function of actual prices, i.e., qd (t) = a − bp(t), b > 0. Price expectation is formed as follows: pe (t + 1) = λp(t) + (1 − λ)pe (t).
- 51. 26 Chaos, Complexity, and Nonlinear Economic Theory Fig. 2.2. Specified relations between expected price and supply. Fig. 2.3. Bifurcation diagram for μ = 0.5, a ∈ [−1.25, 1.25]. It can be shown that under these specifications, the market condition that the demand equals supply is expressed by the following difference equation: pe (t + 1) = (1 − λ)pe (t) + aλ b − λ arctan(μpe (t)) b ≡ f(pe (t)). We will demonstrate the dynamic behavior of the model by simulation. In the remainder of this section, we fix λ = 0.3, b = 0.25 and consider a as a bifurcation parameter for different values of μ. In the case of μ = 0.5, Figure 2.3 depicts the bifurcation diagram for a ∈ [−1.25, 1.25]. We see that there is a unique fixed point for the map f.
- 52. Business Cycles and Chaos with Price Dynamics 27 Fig. 2.4. Bifurcation diagram for μ = 3, a ∈ [−1.25, 1.25]. Fig. 2.5. Bifurcation diagram for μ = 3.5, a ∈ [−1.25, 1.25]. Let us rise the value of μ to 3 and depict the bifurcation diagram with a as the bifurcation parameter as in Figure 2.4. For low values of a, there is a unique fixed point. Around a = −0.9, a period-doubling bifurcation occurs. The stable orbit remains until a reaches 0.9 and then a period- halving occurs. Thereafter, the system settles down again to a unique stable equilibrium. The diagram is symmetrical about the origin because of the characteristic of the arctan function. Figure 2.5 depicts the bifurcation diagram when μ = 3.5. The figure shows a doubling bifurcation into a period-four orbit, which then turns into a period-two orbit and finally a stable equilibrium.
- 53. 28 Chaos, Complexity, and Nonlinear Economic Theory pe* 1.5 1 0.5 −0.5 −0.5 0.5 1 −1 −1 −1.5 Fig. 2.6. Bifurcation diagram for μ = 4, a ∈ [−1.25, 1.25]. Fig. 2.7. Bifurcation diagram for μ = 4.5, a ∈ [−1.25, 1.25]. In Figures 2.6 and 2.7, we depict respectively the bifurcation diagrams when μ = 4 and μ = 4.5. We see that within the period-four orbit, chaos occurs. 2.3 An Inventory Model with Rational Expectations This example discusses a disequilibrium inventory model. This example draws on Hommes (1991: Chapter 28; see also Zhang, 2006b: Chapter 4). Actual labor employed, L(t), is given by the short side of the labor market, L(t) = min{Ld (t), Ls (t)}, where Ld (t) and Ls (t) are respectively demand and supply of labor. Assume Ls (t) = c, where c is constant. Let yd (t)
- 54. Business Cycles and Chaos with Price Dynamics 29 and ys (t) denote respectively aggregate demand and supply. The level of inventories I(t) is positive when there is excess demand, otherwise it is zero, I(t) = max{0, ys (t) − yd (t)}. We denote expected aggregate demand by E(yd (t)) and the desired level of inventories by Id (t). We assume perfect foresight, i.e., E(yd (t)) = yd (t). Suppose Id (t) = βE(yd (t)), where β is a parameter. Production is proportional to labor employed, δL(t). We thus have ys (t) = I(t − 1) + δL(t) yd (t) = E(yd (t)) + Id (t) = (1 + β)E(yd (t)). Setting the demand and supply equal yields the labor demand function: Ld (t) = max 0, (1 + β)E(yd (t)) − I(t − 1) δ . Aggregate demand is assumed to be a linear function of labor employed, yd (t) = a + bL(t). We assume that labor productivity is greater than the marginal propensity to consume, δ b. We have thus completed the description of the model. We now show that the evolution of the system can be described by a difference equation for I. First, we are concerned with L(t) = Ld (t), which implies L(t) ∈ (0, c). From the above conditions, it is straightforward to check that we have L(t) = (1 + β)yd (t) − I(t − 1) δ = (1 + β)(a + bL(t)) − I(t − 1) δ . Solve the above equation for L L(t) = (1 + β)a − I(t − 1) δ − b(1 + β) . Assume δ − b(1 + β) 0. As L(t) ∈ (0, c), then L(t) c is guaranteed. Introduce two parameters: γ1 ≡ (1 + β)a − c(δ − b(1 + β)), γ2 ≡ a(1 + β). It can be shown that I(t) = f(I(t − 1)) is a piecewise function given by I(t) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ I(t − 1) + δc − a − c, I(t − 1) ≤ γ1, −bβI(t − 1) δ − b(1 + β) + aδβ δ − b(1 + β) , γ2 I(t − 1) γ1 I(t − 1) − a, I(t − 1) ≥ γ2.
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- 56. piece of caoutchouc, imitative nostrils of two tin tubes, and imitative lungs in the form of a rectangular wind-chest, he produced with more or less completeness the familiar sounds of n, d, g, k, s, j, v, t, and r. By combining these he produced the words opera, astronomy, etc., and the sentences Vous etes mon ami—Je vous aime de tout mon coeur. By introducing various changes in some such apparatus as this, Professor Willis has developed many remarkable facts concerning the mode in which wind passes through the vocal organs during oral speech. The useful work would be, however, not to imitate vocal sounds by means of mechanism, but to write them so that they should give more information as to their mode of production than our present alphabet affords. Such was the purport of the Phonetic system, which had a life of great activity from ten to twenty years ago, but which has since fallen into comparative obscurity. Mr. Ellis and the Messrs. Pitman published very numerous works, either printed in the phonetic language itself, or intended to develop its principles. Bible Histories, the New Testament, the Sermon on the Mount, Pilgrim's Progress, Paradise Lost, Macbeth, The Tempest—all were printed in the new form; and there were numerous works under such titles as Phonetic or Phonographic Alphabets. Almanacs, Journals, Miscellanies, Hymn-books, Note-books, Primers, Lesson- books, and the like. The intention was not so much to introduce new forms of letters, as new selections of existing letters to convey the proper sounds of words. There was an unfortunate publication, the Fonetik Nuz, which worked more harm than good to the system, seeing that it was made a butt for laughter and ridicule— more formidable to contend against than logical argument. Mr. Bell contemplates something more than this. He has been known in Edinburgh for twenty years in connection with numerous works relating to reading, spelling, articulation, orthoëpy, elocution, the language of the passions, the relations between letters and sounds, logograms for shorthand, and the like. As a writer and teacher on these subjects, he had felt, with many other persons,
- 57. how useful it would be if we could have a system of letters of universal application; letters which, when learned in connection with any one language, could be vocalized with uniformity in every other. There are two obstacles to the attainment of this end: first, that the association between the existing letters and sounds is merely arbitrary; and second, that international uniformity of association is impracticable, because the sounds of different languages, and their mutual relations, have not hitherto been ascertained with exactitude or completeness. Mr. Bell, as he tells us, feeling that all attempted collations of existing alphabets have failed to yield the elements of a complete alphabet, tried in a new direction. Instead of going to languages to discover the elements of utterance, he went to the apparatus of speech itself, endeavoring to classify all the movements of tongue, teeth, lips, palate, etc., concerned in the pronunciation of vocal sounds. By this means, he hoped to obtain, from the physiological basis of speech, an organic scale of sounds which should include all varieties, known and unknown. To transfer these sounds to paper, in the form of visible characters, a new alphabet was necessary. To have adopted letters from the Roman, Greek, or other alphabets, constructed on no common principle of symbolization, would have been to introduce complexity and confusion, and to create a conflict between old and new associations. He therefore discarded old letters and alphabets of every kind. He set himself the task of inventing a new scheme of symbols, each of which should form a definite part of a complete design; insomuch that, if the plan of the alphabet were communicated by diagrams, each letter would teach its own sound, by expressing to the reader's eye the exact position of the sound in the physiological circuit. Could this object be attained, not only would there be a universal alphabet; there would be a scheme of letters representative of sounds, and not, like ordinary alphabets, associated with sounds only by arbitrary conventions.
- 58. Mr. Bell believes that he has achieved this result, and his expositions before the Ethnological Society, the College of Preceptors, and the Society of Arts, have had for their object the presentation of various phases of the system. The fitness of the term visible speech may, be urges, be shown by the analogy of an artist, who, wishing to depict a laughing face, draws the lines of the face as seen under the influence of mirth; be depicts, in fact, visible laughter. Every passion and sentiment, emotion and feeling, has this kind of facial writing; and an idea of it might be expressed on paper by a picture of the muscular arrangements of the face, so that all persons seeing the symbols would have a common knowledge of their meaning. In forming any sound, we adjust the parts of the mouth to certain definite attitudes; and the sound is the necessary result of our putting the mouth in such a shape. If, then, we could represent the various positions of the mouth, we should have in those symbols a representation of the sounds which cannot but result from putting the mouth in the positions symbolized. Now, Mr. Bell claims to have applied this system of symbolization to every possible arrangement of the mouth: he claims that, whatever your language, and whether you speak a refined or a rustic dialect, he can show, in the forms of his new letters, the exact sounds you make use of. If this be so, a Chinaman may read English, or an Englishman Chinese, without any difficulty or uncertainty, after he has learned to form his mouth in accordance with the directions given him by the letters. Nearly all the existing alphabets contain vestiges of a similar relation between letters and sounds—a relation which has nearly disappeared during the changes which alphabetic characters have gradually undergone. Mr. Bell gave the following anecdote illustrating this relation: Shortly before I left Edinburgh, in the early part of last year, an elderly lady called on me, accompanied by two young ladies, who were going out to India as missionaries. The elderly lady had been for upward of twenty years engaged in mission work, and she spoke the language of the district like a native. Nevertheless, she could not teach the English girls to pronounce some of the peculiar sounds which she had acquired by
- 59. habit. They had been for some time under her instruction, but they could not catch the knack of certain characteristic elements. Having heard of 'Visible Speech,' the lady called to solicit my assistance. I know nothing of the language she pronounced before me. Some of the sounds I had never heard in linguistic combinations, though, of course, I am acquainted with them theoretically. I saw the young ladies for half an hour, but this proved long enough to give them the power of pronouncing the difficult sounds which, while they did not know precisely what to do, they could not articulate. Strangely enough, since I came to reside in London, I heard a clergyman and former missionary, speaking of these very girls, remark on the great success with which they pronounced the Canarese language before they left this country; and the speaker knew nothing of their previous difficulty, or how it had been overcome. The system analyzes all sounds according to the mode in which they are produced. The number of sounds discriminated in various languages amounts to several times the number of letters in the English alphabet; and even in English, although there are only twenty-six letters, there are at least forty different sounds. The Church Missionary Society employ nearly two hundred different letters or symbols in their several printed books; and the list is even then imperfect as regards many of the languages. Mr. Bell finds thirty symbols sufficient to denote all the two hundred varieties of vowel and consonant sounds. What kind of symbols they are, we do not know, (for a reason presently to be explained;) but he states that, while each elementary sound has its own single type to express it in printing, he requires only thirty actual types to express them as used in language. Each symbol has a name, which does not include the sound of the letter, but merely describes its form. The learner has thus at first only to recognize pictures. But the name of the symbol also expresses the arrangement of the mouth which produces the sound; so that, when the symbol is named, the organic formation of its sound is named at the same time. In order that thirty symbols may denote two hundred sounds,
- 60. Mr. Bell has adopted certain modes of classification. All vowels receive a common generic symbol, all consonants another; vocality and whisper have their respective symbols; so have inspiration, retention, and expulsion of breath; so have the touching and the vibration of the several vocal organs; so have the lips, the palate, the pharynx, the glottis, and the different parts of the tongue; so has the breathing of sounds through the nostrils, or through nearly closed teeth. There are thirty of these generic meanings altogether, and they are combined to make up letters, every part of every letter having a meaning. The thirty symbols need not be represented mechanically by exactly thirty types; they may be embodied in a larger or smaller number, according to taste or convenience; such of the symbols as together represent simple elements of speech being properly combined in single types. The highest possible advantages of the system, we are told, would be secured by extending the number of types to about sixty. At present, I and my sons—as yet the only experts in the use of visible speech—write the alphabet in a form that would be cast on between forty and fifty types, which is but little more than the number in an ordinary English fount, including diphthongs and accented letters. This number does not require to be exceeded in order to print, with typographic simplicity, the myriad dialects of all nations. Mr. Bell pointed out the prospective usefulness of his system in telegraphic communication. The symbols of speech may, in all their varieties, be transmitted by telegraph, through any country, without the necessity for a knowledge of the language adopted on the part of the signaller. He would only have to discriminate forms of letters; he may be totally ignorant of the value of a single letter, and yet may convey the telegram so as to be intelligible to the person to whom it is virtually addressed. It is known that the telegrams from India now reach London in a sadly mutilated and unintelligible state, owing to their passing through the hands of Turkish and Persian agents who do not know the English alphabet; an evil
- 61. which, it is contended, would be removed by the adoption of the new system. The mode in which Mr. Bell illustrated his method was curious and interesting. His son uttered a great variety of sounds—whispered consonants, vocal consonants, vowels, diphthongs, nasal vowels, interjections, inarticulate sounds, animal sounds, mechanical sounds —all of which are susceptible of being represented in printed or written symbols. Then, the son being out of the room, several gentlemen came forward and repeated short sentences to Mr. Bell, some in Arabic, some in Persian, some in Bengali, some in Negro patois, some in Gaelic, some in Lowland Scotch, some in Norfolk dialect; Mr. Bell wrote down the sounds as he heard them, without, except in one or two cases, knowing the purport of the words. The son was called in, and, looking attentively at the writing, repeated the sentences with an accuracy of sound and intonation which seemed to strike those who were best able to judge as being very remarkable. There is something a little tantalizing in the present state of the subject. We know that there is a system of symbols, but we do not know the symbols themselves. Mr. Bell states that, besides the members of his own family, only three persons have been made acquainted with the symbols, and the details of their formation— namely, Sir David Brewster, Professor de Morgan, and Mr. Ellis. He has not intended, and does not intend, to secure his system to himself by any kind of patent or copyright; and yet, if he made it fully public at once, he would lose any legitimate hold over it to which he is rightly entitled. He has submitted his plan to certain government departments, but has found that it is nobody's business to take up a subject which is not included in any definite sphere of duty. He has next endeavored to interest scientific societies in the matter, so far as to induce them to urge the trial of his plan by the government. He says: I am willing to surrender my private rights in the invention pro bono publico, on the simple condition that the costs of so introducing the system may be
- 62. undertaken at the public charge. Teachers there must be, because the publication of the theory of the system and the scheme of symbols must necessarily be supplemented by oral teaching of the scales of sound, in order that the invention may be applied with uniformity. The reading of the paper gave rise to some discussion at the Society of Arts, not as to the value and merit of the system itself, but as to anything which the society can do in the matter. It is one rule of the society that no new invention shall be brought forward without a full explanation of the modus operandi as well as of the leading principles; and in this case, the objection lay that the inventor declined to make public, unless under some government agreement, the actual secret of his method. Mr. Bell replied that, if even he were to write a sentence in view of the audience, it would add very little to their real knowledge of the subject; but he furthermore said he was ready to explain the details of the system to any committee whom the council of the society, or any other scientific body, may appoint. To us it appears that neither Mr. Bell nor the society is open to blame in the matter. He has the right to name the conditions under which he will make his system public; while they have the right to lay down rules for the governance of their own proceedings. The results actually produced struck the auditors generally with surprise; and there can be little doubt that the system will in some way or other, at all events, work itself into public notice.
- 63. Comparative Mortality of Great Capitals. Our recent alarm at the appearance and progress of the cholera in London may have drawn the attention of many who had before been accustomed to pass them by with indifference, to those columns in the papers in which the reports of the Registrar-General on the state of the public health are from time to time recorded. But we are perhaps hardly yet sufficiently awake to the importance and interest of the statistics there contained, any more than to the value of the short and, at first sight, rather unintelligible tables which embody, day after day, the meteorological phenomenon collected in London from so many different points on our own coast and those of adjacent countries. These last statistics have an interest which does not yet belong to those which relate to the public health, in that they embrace reports from so many distinct places which can be compared together. We, of course, only publish our own statistics of health, disease, births, and deaths; and we have not yet seen our way to the information that might be gathered by a comparison of our own condition in these respects with that of others under similar circumstances. The interest and value of such a comparison is obvious enough; and some of the results which might be hoped from it, if it were systematically and scientifically made, may be guessed at by the perusal of a thin volume of less than two hundred pages, lately published in Paris by M. Vacher, [Footnote 136] which at first sight may seem not to promise very much except to professional readers, but from which we shall take the liberty of drawing a few facts which certainly seem worthy of the attention of the more general public. [Footnote 136: Etude Médicale et Statistique sur la Mortalité à Paris, à Londres, à Vienne et à New-York en 1865. D'aprés les Documens officiels, avec une Carte Météorologique et Mortuaire. Par le docteur L. Vacher. Paris: F. Savy, 1866.]
- 64. Canning once said, in answer to some one who alleged a well- known fact against him, that there was but one thing more fallacious than a fact, and that was a figure. We must all be ready to allow that the results which we see embodied so neatly in a set of figures in statistical tables are, after all, but approaches to the truth; and they are not put forward as anything more. Still, there is often a wonderful accuracy about the average results given by statistical inquiries; and it is obvious that when the result of one calculation is confirmed by that of another independent of the former, or when one uniform result is given by a continued series of inquiries, or when there is a very decided preponderance on one side of a comparison, such as cannot be accounted for by chance, it would be absurd to refuse to assent to conclusions thus obtained. With this single preliminary remark, let us proceed to some of the facts collected for us by M. Vacher. He begins by giving due credit to this country for having taken the lead in the publication of the kind of statistics with which he has to deal. The reports of the Registrar-General are all that he can desire. New York and Vienna have followed, more or less fully, the example set in London. It has also been copied in St. Petersburg, as far as the registration of deaths is concerned; and it is hoped that a weekly publication of the results will soon be made in that city. Paris joined the movement at the end of 1864 or the beginning of 1865. There is, however, some difference of system. The chief point is, that in England the medical man who attends a sick person reports the cause of death; in Paris there are certain official physicians, vérificateurs des décès, and these, instead of the attending physician, assign the cause. The superiority of the English system seems to be acknowledged. M. Vacher's book is founded on the reports thus produced. His first business is, of course, to settle approximately the population of the four capitals with whose statistics he deals—a matter of considerable difficulty, even with all the results of the census before him. He calculates the number of the inhabitants of
- 65. Paris in 1865 at 1,863,000; those of London were 3,028,600; those of Vienna, 560,000; and those of New York, 1,025,000, (in 1864.) At the present rate of increase, Paris will double its population in 32 years, London in 40, Vienna in 44, and New York in 13½. On the other hand, this increase is not to be set down to the excess of births over deaths, which in London, in 20 years before 1861, was only 328,189—about a third of the actual increase, (35 per cent.) In a similar period, the births exceed the deaths in Paris by only 13 (and a fraction) per cent of the whole increase. Immigration has therefore the largest share in the increase of the population. A flow is continually setting in from the country to the town in the age in which we live, and it enriches the largest towns, and the capitals especially. New York, receiving annually so many immigrants from Europe, is, of course, beyond the others in its gains from this source. Paris has undergone great vicissitudes as to the number of its inhabitants. In 1762, the population seems to have been about 600,000. It fell off immensely during the Revolution; even in 1800 it was only 547,756. From 1790 to 1810 the number of deaths exceeded the number of births. Since that time the proportion has been reversed, except in years of great epidemics. Of the four capitals with which M. Vacher deals, Vienna, the smallest, had the largest proportion of deaths in 1865. In Vienna the proportion was 1 to 31 of the inhabitants; in Paris, notwithstanding the ravages of the cholera in October—causing 6591 deaths (nearly an eighth of the whole)—it was 1 to 36; in New York, 1 to 40; in London, 1 to 41. In Paris, London, and New York, the death rate has diminished in its proportion to the population for some time past. In Paris, in the three decades of years from 1830 to 1860, it fell successively from 1 to 31, to 1 to 34, and then to 1 to 38. There has been the same improvement in the other two cities. In New York, fifteen years ago, the rate of deaths was 1 to 22—nearly twice as high as at present. We do not see any statement in M. Vacher's pages as to the case of Vienna. He attributes the improvement in Paris to some extent to the great public works and measures for securing the health of the
- 66. population which have marked the second empire; but much more, it would seem, to the better management of the hospitals. In Paris and Vienna a much larger proportion of the inhabitants die in hospitals than in New York and London; and, as far as we are concerned, M. Vacher includes workhouses and asylums of all kinds under the general name of hospitals. He finds, on comparing some scanty statistics of the last century with the facts of the present, that in old times the number of deaths in hospitals was far greater in proportion to the cases admitted than now; and he thinks that, in Paris at least, this almost explains the improvement in the death- rate. In New York the same improvement may have had many causes, but it is remarkably coincident as to time with the magnificent changes made, at an immense cost, in the water supply of that city. From some meteorological tables compiled with great care by M. Vacher, we gather the rather surprising result that the variations of temperature during the year, which have considerable influence on the death-rate, are greatest at Vienna, (nearly 27°,) next at New York, (25°,) much lower in Paris (17°,) and lowest of all in London, (15°.) One of the most interesting questions at the present time on this subject is that of the water supply. M. Vacher begins with a cordial tribute to the Romans on this head. The magnificent aqueducts by which the city of Rome was supplied date from the time of the early republic, though the emperors increased their number. At an early point of their history, therefore, the Romans were wise and liberal enough to dispense with the waters of the Tiber for drinking. They carried their system everywhere when they became the masters of the world; in France, in Spain, and in Italy many aqueducts can still be traced which were their work. We may be quite certain that if Britain were now a Roman province, the Thames water companies would never be allowed to supply water except for the streets, and great aqueducts would long since have brought us the pure water of Bala Lake or Windermere. Thanks to the popes, modern Rome though not so profusely supplied as in imperial times, is still very far in advance of all other cities in the
- 67. world in this respect. [Footnote 137] M. Vacher reckons the water supply in ancient Rome as 1492 litres a day for each inhabitant; in modern Rome it is 1040; in New York, 159; in Vienna. 134; [Footnote 138] in Paris, according to the new system, 109; in London, 132. But no city seems to have its houses so well supplied as London; in Rome a great quantity of water is wasted, being left to run away from the fountains, while the houses are not conveniently provided with water. We suppose that our old friend the house-cistern, against which we have heard so many complaints lately, is not an essential feature in our system of house supply. [Footnote 137: M. Vacher attributes the salubrity of Rome—for, considering its position, it enjoys remarkable salubrity—to the abundance and good quality of its water. Lancisi, who practiced there as a physician in the last century, accounts for the longevity of its inhabitants in the same way. At all events, remarks M. Vacher, il est impossible de n'étre pas frappé de ce fait, que les historiens ne mentionnent pas un seul example de peste à Rome, et qu'au moyen age et dans les temps modernes elle a constainment échappé aux atteintes de la pests et du choléra, qui ont sévi à plusteurs reprises en Italie. But Rome has certainly been visited by the cholera more than once, and the rest of the statement is surely contrary to history.] [Footnote 138: This statement is, however, an anticipation. The municipality of the Vienna has undertaken some immense works in order to improve the water supply, at a cost of 16,000,000 florins. The works are not yet completed: but M. Vacher gives the quantity of water for each inhabitant which they are expected to furnish. Hitherto the city has been supplied, it would seem, partly from the Danube, partly by wells.
- 68. The new supply will be drawn from three different sources among the neighboring mountains.] M. Vacher gives the following conclusions as to the sanitary effect of good and abundant water. He tells us that inorganic substances contained in water are comparatively innocuous to the health of those who drink it; on the other hand, great injury is caused by the presence of organic matter. The best water in Paris—that of the springs on the north—contains nine times as much of calcareous salts as the water of the Seine; but it is justly preferred for drinking purposes. On the other hand, M. Vacher quotes the testimony of M. Bouchut, a professor at the Ecole de Médecine, for the fact that he noticed the frequency of epidemic diarrhoea during the summer months in the Quartier de Sèvres and that it had been almost stopped in cases where the doctors had ordered the water of the Seine to be no longer used, and had substituted for it water from the artesian well of Grenelle. He adds his own experience at the Lycée Napoleon, which is supplied from the reservoir of the Pantheon, which receives its water from the Seine and the aqueduct d' Arcueil. He had known as many as fifteen students at once ill of diarrhoea, and the disease was stopped by the alcoholization of all the water. [Footnote 139] [Footnote 139: P, 106. M. Vacher here cites the Indian case quoted by Mr. Farre in is cholera report. The natives in India drink boiled water as a preventative against cholera; and it has been found that out of a great number in the family of a single proprietor in Calcutta, all of whom took this precaution, not a single person had been attacked even in the worst times of the prevalence of cholera. But Dr. Frank has disapproved at least the universality of this fact.] As regards cholera, the proof is even more striking than that lately furnished in the case of London by the great and almost exclusive ravages of that disease in the eastern districts. Mortality by cholera
- 69. seems ordinarily, as M. Vacher tells us, to follow the laws of general mortality, that is, it prevails most in those districts which are ordinarily the most unhealthy. But the one element of good or bad water supply seems to be enough to counterbalance the influence of the other causes which affect the comparative mortality of districts. For instance, difference of elevation is supposed to be one of these causes. Mr. Farre tells us that the mortality of a district is in inverse proportion to the elevation: that in nineteen high districts the proportion of deaths by cholera was as 33 to 10,000; in the same number of low districts, as 100 to 10,000. This law, however, is not enough, nor is it free from exception. Sometimes places loftily situated are attacked and lower places are spared. The elevation of Montmartre is almost equal to that of Belleville; but Montmartre had last year 3.6 cholera cases to 1000, Belleville only 1.1. Again, a rich quarter has ordinarily immense advantages over a poor quarter. The mean mortality by cholera in the poorer arrondissements of Paris was almost three times as great as that in the rich arrondissements. The reason is obvious: the poor work hard, have insufficient food, and are crowded together in discomfort and want; the rich are well fed, not overworked, well and healthily housed. Yet there was one arrondissement of Paris, and that one of the very poorest, which in the three first visitations of cholera (1832, 1849, 1854) had actually the lowest proportion of deaths by cholera of all these districts. In 1865, it had barely more deaths than the very richest of all, that of the Opéra, which headed the list on that occasion as the most lightly visited. This arrondissement was Belleville. Another cause of comparatively greater mortality is density of population; but here again we are met by the fact that this fortunate Belleville is very densely populated. The nature of the soil is another. M. Vacher mentions a number of departments in the centre of France which have never yet been attacked by cholera. They are those which consist of a huge granitic mass, like an island in the midst of the more recent formations around them. Nevertheless, though this will explain much, and though Belleville has an advantage in this respect over many of the arrondissements of Paris, still it has the same
- 70. geological formation as Montmartre, which had three times as many deaths (in proportion) from cholera. In short, there is no way left of accounting for its comparative exemption, except that which we have already mentioned, the superior character of the water consumed by its inhabitants. The argument certainly seems as complete as it can possibly be, and we know that it has been strongly confirmed by our own late experience. Let us hope that no time may be lost in acting on the lesson which we have received. We pass over some interesting statements on the meteorological phenomena which were observed during the prevalence of the cholera last year in Paris. [Footnote 140] [Footnote 140: M. Vacher here tells a story of his endeavor to make some ozonometrical observations in the Paris hospitals, which were prohibited by the Directeur de l'Assistance publique—an officer of whom M. Vacher is continually complaining on the ground that they would frighten the patients. He remarks that on one occasion when travelling in the pontifical states, some gendarmes found in his possession a psychrometer and an aneroid barometer, and thought they were weapons of destruction. He would have been arrested but for M. Matteucci, then Director of Police. He complains bitterly of the comparative want of enlightenment in the administration of his own country. But no hospital would have allowed his experiments.] M. Vacher rather contradicts current opinion by some remarks he has made as to the relation of cholera to other diseases. Sydenham has remarked that when several epidemic diseases are rife during the same season, one of them usually absorbs to itself, as it were, the bulk of the mortality, diminishing the influence of the rest even below the ordinary level. Thus in the year of the great plague in London, just two centuries ago, the smallpox was fatal to only thirty-eight persons, its average being about eleven hundred.
- 71. However, the general fact is now questioned. In October last, though 4653 persons were carried off by cholera, the mortality by other diseases in Paris was greater than in any other month of the year. Yet October is usually one of the most healthy of all the months; and the epidemic maladies which ordinarily rage during the autumn—typhoid fever, small-pox, diphtheria, croup, whooping- cough, erysipelas, and puerperal fever—were prevalent to an extraordinary degree. It is curious also that there was an unusual number of children born dead. The most destructive of all ordinary complaints is undoubtedly consumption. At Vienna it actually causes 25 per cent of the deaths, at Paris 16 per cent, at London nearly 12 per cent, at New York 14 per cent. It is more frequent in women than men; it is twice as destructive in poor quarters as in rich quarters; the age which suffers most from it is between 25 and 40. The difference between the sexes M. Vacher attributes to the more confined and retired life led by women. If observations in Paris are to be taken as enough to furnish a general conclusion, it would appear that more consumptive patients die in the spring than in the autumn. Here again a common opinion is overthrown. The most destructive months are March, April, and May: the least destructive are September, October, and November. We believe that in this country the fewest consumptive patients die in winter, and the most in summer. M. Vacher also attacks the notion that maritime climates are the best for consumptive cases. New York is situated on the sea, but it loses as many by consumption as London; and in the maritime counties of Kent, Sussex, Hampshire, Dorset, and Devonshire, the deaths by consumption are as 1 in 7 of the whole; while in the Midland counties of Warwickshire, Buckinghamshire, Worcestershire, and Oxfordshire, they are as 1 in 9. Les phthisiques qu'on envoie à Nice et à Cannes, ou même sur les bords du Nil, sur la foi d'un passage de Celse, y meurent comme ceux qui restent sous le ciel natal. Ceux-la, seuls en reviennent guéris, chez qui le mal n'était pas sans ressources et qui auraient guéri partout ailleurs, (p. 129.) We must remember, however, that
- 72. if such patients are sent to the seaside, and die there, they raise the death-rate there unfairly. M. Vacher insists that the guiding principle in selecting a place for the residence of a consumptive patient should be the absence of great variations in the temperature rather than the actual number of deaths by the disease. Consumption, he says, is unknown in Iceland; but that is not a reason for sending a consumptive patient to that island. As to New York, we have already quoted his observation as to the variableness of the temperature there, notwithstanding its maritime position. Although we have already stated the results of a general comparison of the mortality in the four capitals—results very favorable to the salubrity of London—it may be interesting to our readers to learn the state of the case with regard to particular classes of disease. In most cases, of course, we have the list in actual numbers: our comparative immunity is only evident when the great excess of our population is considered. In zymotic diseases we have little more than a majority of a thousand over Paris; but then we must remember that in the year of which M. Vacher speaks between 5000 and 6000 persons in Paris died of cholera. This, therefore, would seem to be one of the classes of disease as to which we are really worst off. As to constitutional diseases, consumption, cancer, scrofula, gout, rheumatism, and others, Paris exceeds us in proportion; and it is the same with diseases of the nervous system. From diseases of the heart we lose between two and three times as many as the Parisians; this proportion, therefore, is greatly against us. On the other hand, in diseases of the digestive organs, Paris, notwithstanding its inferior population, exceeded London by a hundred deaths in the last year. London, however, regains a sad preeminence when we come to diseases of the respiratory organs, asthma, bronchitis, influenza, and the like: Paris losing between 7000 and 8000 a year against our 12,500. It is in the commoner diseases that the worst features of London mortality in 1865 were found. Typhoid was nearly three times as fatal last year in London as in Paris; measles four times as fatal;
- 73. scarlatina not far short of twenty times; whooping-cough more than thirteen times. As the population of London is to that of Paris as five to three, it is clear to how great an extent the balance was against us. It was probably an accident. These diseases prevail very generally for a time, and then retire: and we have lately been visited by a period of their prevalence. We have hitherto spoken only of diseases; but M. Vacher's researches extend to the comparative frequency of deaths of other kinds. In suicides, New York has the best account to give, Paris the worst. To speak roughly, London has twice as many suicides as New York, Vienna twice as many as London, Paris more than twice as many as Vienna—in comparison, that is, with the total number of deaths of all kinds. The actual numbers stand thus: Paris 716, London 267, Vienna 813, New York 36. For the last nine years there has been little change in the number in London; in New York it has diminished, in Paris it has increased, having more than doubled itself since 1839. The two years, 1848 and 1830, which were marked by revolutionary movements, were also marked by a diminution in the number of suicides. The relative proportion of suicides increases with age; that is, it is four times as frequent with people above 70 as with people between 20 and 30. Paris has for a long time been noted as a city in which there were more suicides than any other. More than eighty years ago, Mercier noted this, and attributed it to the rage for speculation. Other writers have since attempted to find a reason for it in the prevalence of democratic ideas. We suppose that both democratic ideas and speculation are not unknown in New York, yet that city (and indeed the State itself) is remarkably free from suicides, and a great number of those that occur are said to be of Europeans. But if Paris bears the palm in self-slaughter, no city can vie with London in slaughter of another kind. Violent deaths are nearly three times as frequent in London as in Paris. As many as 2241 persons were slain in London last year; as many, that is, as would be enough for the number of the killed in a sanguinary battle: 328
- 74. were burnt, 405 were suffocated, (this probably includes children overlaid by their mothers,) 40 were poisoned, 767 disposed of by fractures and contusions, 232 were killed by carriage accidents; leaving 469 to be laid to the account of other accidents. In the other three capitals the proportion of deaths by accidents to the whole number of deaths ranges from under one per cent to under two per cent; in London it is just three per cent. Finally, London had 132 murders to give an account of in 1865, Paris had 10, and New York only 5. We are sorry that the last fact which we glean from M. Vacher's interesting tables must be one rather disparaging to the great Transatlantic city which we have last named. Disparaging, that is, positively rather than comparatively; and we fear that, if the statistics which we are now to quote do not reveal a terrible state of things in London also, it is because on this head our admirable system of registration has given M. Vacher no assistance at all. Quant à la ville de Londres, he says, il m'a été impossible d'arriver à connaitre le chiffre de ses mort-nés. Le Bulletin des Naissances et des Morts ne donne d'ailleurs aucun renseignement à ce sujet. He expresses his opinion that, if the numbers were given, London would have quite as bad a tale to tell as Paris or New York. But the figures in these cities are sufficiently startling. In Paris the children born dead are to the whole number of deaths as one to ten; in New York as one to fifteen; in Vienna they are as one to twenty-three. Twenty years ago, the Préfet of the Seine addressed a circular to the maires of Paris, in which he drew their attention to the great number of these children, and pointed out that it was natural to conclude that their deaths were too often the result of crime. In New York similar complaints have been made, and we are significantly told that full reports cannot be obtained on the subject. As to London, we find a large number of deaths, 1400 or 1500 a year, set down to premature birth and debility. We fear it would be quite impossible to give an account of the number of births which are prevented—contrary to the laws of God and man alike. We need hardly do more than allude to the frightful increase of
- 75. infanticide, on which Dr. Lankester has lately spoken so strongly. Mr. Humble's Essay on the subject in Mr. Orby Shipley's volume contains some very startling statistics. There are as many as 12,000 women in London to whom this crime may be imputed. In other words, says Mr. Humble, one in every thirty women (I presume, between fifteen and forty-five) is a murderess. We must hope that there is exaggeration about this; but if it were one in every thirty thousand, it would be bad enough—a state of things calling down the judgments of heaven on the land. The Anglican writer to whom we have just alluded speaks with some apparent prejudice against the most obvious remedy for infanticide—the establishment of foundling hospitals, perfectly free. There may be some objections to these institutions, but we must confess that, in the face of the facts on which we are commenting, they seem to us rather like arguments against life-boats because they may encourage oversecurity in exposure to the dangers of the sea. If Mr. Humble will read, or read again, Dr. Burke Ryan's Essay on Infanticide, which gained the Fothergillian prize medal some time ago, and in which the fact seems to be proved that the crime is more common in England than anywhere else, he will perhaps see reason to conclude, from the French statistics there adduced, that foundling hospitals are more effectual in preventing this abominable evil than anything else that has ever been devised.
- 76. Miscellany. New Electric Machines.—At the conversazione given by the president of the Royal Society at Burlington House, London, the display of newly constructed astronomical, optical, and other philosophical instruments afforded a gratifying proof of improvements in the mode of construction, and of increased skill on the part of the constructors. The large spectroscope, which is to be used in combination with Lord Rosse's monster telescope, was a triumph of workmanship and of philosophical adaptation of means to ends; and we may expect ere long to hear of important discoveries in spectroscopic phenomena. Mr. C. W. Siemens and Professor Wheatstone exhibited each one a remarkable electric machine of his own invention, which demonstrated in a surprising way the convertibility of mechanical force into electricity. In these machines, a bar of soft iron, wrapped lengthwise in copper wire, is made to rotate between two other bars of soft iron, which are fixed. The rotating bar is inoculated, so to speak, with a small touch of magnetism, and then being set spinning very rapidly, the small touch is generated into a stream of electricity, which passes off with a crackling noise, increasing or diminishing in proportion to the rotation. In a laboratory, such a machine would be highly serviceable, as it could be used to generate large quantities of electricity very cheaply, and there is no doubt but that many other ways of turning it to account will be discovered. Mr. Siemens has already discovered one most important way, namely, the lighting-up of buoys and beacons at a distance from the shore, by sending a current of electricity to them through a submarine cable. That is the way in which he purposes to employ the electricity generated by his machine: his method has been approved by the Commissioners of Northern Light-houses, who intend to apply it to light the buoys and beacons that mark the most dangerous spots round the coast of Scotland. But of all wonderful electric machines,
- 77. the one invented by Mr. H. Wilde of Manchester is the most wonderful. A machine which weighs about four and a half tons, including one ton of copper wire, and which requires an eight-horse steam-engine to keep its armature in rotation, must necessarily produce tremendous effects. It gives off electric fire in torrents: the light produced is intense, and is quite as useful to photographers as sunlight, with the advantage over the sun, that it can be used on dark days and at night. This light, as we hear, is already employed in manufacturing establishments, and is to be introduced into light- houses. A French company, who have purchased the right to use it in France, will try it first in the light-house on Cape Grisnez, whence, as is said, the light will radiate not only all across the Channel, but some distance into the southern counties of England. Besides the production of light, the new machine is applicable to important manufacturing purposes; the size of the machine being altered to suit special circumstances. A well-known firm at Birmingham are about to use it, instead of a galvanic battery, for the deposition of copper on articles required to be coated with that metal. In this case, the electricity of the machine is substituted for the acid and zinc of the battery, and will cost less. In another instance, the machine is to be used for the production of ozone in large quantities for employment in bleaching operations. Professor Tyndall exhibited the sensitive flame, on which he had given a lecture at the Royal Institution: or, to be more explicit, he made experiments to show the action of sound on flame. The results are remarkable. A tall flame, looking like an ordinary gas-flame, issuing from a circular orifice in an iron nipple, behaves in an extraordinary way when, by increased pressure, it is raised to fourteen or sixteen inches in length. If a shrill whistle be blown in any part of the room, it suddenly drops down to about half the length, and rises again immediately on cessation of the sound. A blow of a hammer on a board produces a similar effect; and still more so when the blow is on an anvil: the flame then jumps with surprising briskness, the reason being that the ring of the anvil combines those higher tones to which the flame is most sensitive. So tuning-forks, at the ordinary pitch, produce no effect; but if made to vibrate one
- 78. thousand six hundred, or two thousand, or more times in a second, the flame responds energetically. In another experiment, a fiddle is played in presence of a flame twenty inches in length—the low notes produce no effect; but when the highest string is sounded, the jet, to quote Professor Tyndall's own words, instantly squats down to a tumultuous bushy flame, eight inches long. And the same effect is produced by strokes on a bell at twenty yards' distance: at every stroke the flame drops instantaneously. This last experiment is a good illustration of the rapidity with which sound is propagated through air, for there is no sensible interval between the bell-stroke and the shortening of the flame. Another flame, nearly twenty inches long, is yet more sensitive, for the rustle of a silk dress, a step on the floor, creaking of boots, dropping of a small coin, all make it drop down suddenly to eight inches, or become violently agitated. At twenty yards' distance, the rattle of a bunch of keys in the hand shortens the flame, and it is affected even by the fall of a piece of paper, or the plashing of a raindrop. To the vowel U, it makes no response; to O, it shakes; E makes it flutter strongly; and S breaks it up into a tumultuous mass. Many more instances might be given, but these will suffice to show that surprising effects are produced by sound. To the scientific inquirer they will be serviceable as fresh illustrations in the science of acoustics. Chambers's Journal.
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