Table Of ContentHIERARCIUCALLY STRUCTURED ECONOMIES
THEORY AND DECISION LIBRARY
General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich)
Series A: Philosophy and Methodology of the Social Sciences
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Series C: Game Theory, Mathematical Programming and Operations Research
Series D: System Theory, Knowledge Engineering and Problem Solving
SERIES C: GAME THEORY, MATHEMATICAL PROGRAMMING
AND OPERATIONS RESEARCH
VOLUME 15
Editor: S. H. Tijs (University of Tilburg); Editorial Board: E.E.C. van Damme (Tilburg),
H. Keiding (Copenhagen), J.-F. Mertens (Louvain-la-Neuve), H. Moulin (Durham), S. Muto
(Tohoku University), T. Parthasarathy (New Delhi), B. Peleg (Jerusalem), H. Peters
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Scope: Particular attention is paid in this series to game theory and operations research, their
formal aspects and their applications to economic, political and social sciences as well as to
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methods to individual and social decision making.
The titles published in this series are listed at the end of this volume.
HIERARCHICALLY STRUCTURED
ECONOMIES
Models with Bilateral Exchange
Institutions
by
WILLY SPANJERS
University of Saarland, Saarbriicken, Germany
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-1-4419-4 771-0 ISBN 978-1-4757-5394-3 (eBook)
DOI 10.1007/978-1-4757-5394-3
Printed on acid-free paper
All Rights Reserved
© 1997 Springer Science+B usiness Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any fonn or by any means, electronic or mechanical,
including photocopying, recording or by any infonnation storage and
retrieval system, without written pennission from the copyright owner.
CONTENTS
ACKNOWLEDGEMENTS vii
1 INTRODUCTION 1
2 HIERARCHICALLY STRUCTURED
EXCHANGE ECONOMIES 9
2.1 Hierarchical Structures 10
2.2 Institutional Characteristics 13
2.3 The Model 18
3 MONO PRICING IN HIERARCHICAL
TREES 23
3.1 The Model 24
3.2 Existence of Equilibrium 28
3.3 Some Examples 39
4 BID AND ASK PRICES IN HIERARCHICAL
TREES 49
4.1 The Model 50
4.2 Existence of Equilibrium 55
4.3 Three Examples 66
4.4 A Dynamic Interpretation of the Exchange Process 73
4.5 Endogenous Structures of Trade Relationships 76
4.6 An Example with Choice of Trade Partners 83
5 ARBITRAGE 87
5.1 The Model 89
v
Vl HIERARCHICALLY STRUCTURED ECONOMIES
5.2 Existence Results 99
5.3 Results on Market Structures 110
5.4 An Example with Arbitrage 114
6 HIERARCHIES 123
6.1 The Model 124
6.2 Some Results 130
6.3 Some Examples 134
A MATHEMATICAL APPENDIX 145
A.1 Terminology on Graphs 145
A.2 Optimization 149
B EQUILIBRIUM CONCEPTS 153
B.1 Subgame Perfect Equilibrium 153
B.2 Equilibrium in Very Nice Plays 158
B.3 Approximate Equilibrium in Very Nice Plays 163
REFERENCES 169
INDEX OF AUTHORS 173
INDEX OF SUBJECTS 175
ACKNOWLEDGEMENTS
The research for this monograph was carried out over the last eight years, at
a number of institutions. At the Department of Econometrics at Tilburg Uni
versity, the Netherlands, it was part of the AIO-Project "Social Institutions in
General Equilibrium" from 1988 to 1992. As part of the sub program "General
Equilibrium Theory and Microeconomic Policy" of the VF-Program "Com
petition and Cooperation" the project was supervised by Prof. Dr. P.H.M.
Ruys. From 1992 to 1994 this research continued at the Institute of Mathe
matical Economics (IMW) at the University of Bielefeld, Germany, as part of
DFG-Project TR-120/4-2 "Incentives in Markets and Institutions" which was
supervised by Prof. Dr. W. Trockel. This monograph was finally writen at the
University of Saarland, Saarbriicken, Germany in the years 1995 and 1996.
The research, and therefore the monograph, benefited greatly from the stim
ulating discussions and helpful comments at these institutions, at workshops
and at conferences, where I presented parts of the research. The support of a
large number of friends and colleagues, too many to name in person, is grate
fully acknowledged. Special thanks, however, deserve Prof. Dr. P.H.M. Ruys
for showing me how to conduct innovative research, Prof. Dr. W. Trockel for
teaching me mathematical rigor, and Prof. Dr. J. Eichberger for convincing
me a well-chosen example can say more than a thousand words. Shortcomings
in these areas are those of the apprentice, not of his masters. Helpful remarks
and careful reading from Dr. P.J.J. Herings of an earlier draft improved the
final manuscript significantly. Any remaining errors are mine. Final thanks go
to H.A. Admiraal who skillfully turned my sketches into figures.
vii
1
INTRODUCTION
The problem of allocating scarce resources is the basic problem of economics.
In 1776, Adam Smith suggested that for certain goods competitive markets al
locate scarce resources to the well-being of all, an "invisible hand" coordinating
the ruthless pursuit of self interest by the participants in the market.
This notion of Smith's has been translated into formal models of competitive
markets in, e.g., Walras (1874) and Debreu (1959). They model competitive
markets as markets in which all trade takes place at a single vector of prices
which holds for all market participants, and which is assumed by each of them
not to be influenced by his trades on the market. As stated in the First Theorem
of Welfare Economics, in economies with a complete system of interconnected
competitive markets, every equilibrium allocation is Pareto efficient. Thus, in
the context of these models, Pareto efficiency can be interpreted as describing
the notion of "the well-being of all".
This result is at the core of general equilibrium theory. General equilibrium
theory typically analyzes economies with a Walrasian market, trade processes
that lead to an equilibrium in this market, and models without markets that
yield Walrasian allocations.1 Essentially, it focuses on one particular type of
market which is analyzed in great depth and with enormous rigor, often by
means of sophisticated mathematical techniques.
To some extend industrial organization is a direct opposite to general equilib
rium theory. In the realm of industrial organization, a large number of different
1 For an introduction to general equilibrium theory, see e.g. Hildenbrand and Kirman
(1988)
1
2 CHAPTER 1
market forms is analyzed, the analysis typical lacking the depth and rigor that
is characteristic for general equilibrium theory. More often than not, industrial
organization seems to be a field that is anecdotal and ad hoc in its very na
ture. Be this as it may, in industrial organization a lot of interesting problems
are successfully addressed that are difficult to analyze in the mathematically
more sophisticated, but for this purpose too little flexible, models of general
equilibrium theory. 2
One of these interesting economic problems is about the development of the
structure of markets and, in general, economies and societies. 3 The problems
of comparing the institutional structures of markets and endogenizing them
seems to be at the heart of industrial organization. This is reflected in, e.g.,
theories on product differentiation, theories on market entry and theories on
horizontal and vertical integration. Endogenizing a part of the environment in
which the agents operate is central in all those theories.
A number of scholars have worked on connecting general equilibrium theory
and industrial organization in building models that integrate non-competitive
markets in the general equilibrium framework. Negishi (1961) introduces a
general equilibrium model that allows for both competitive markets and mo
nopolistic competition. In such models, agents act on (some of) the partial
markets that are present in the economy. These partial markets can be in
terpreted as exchange institutions with respect to which the agents coordinate
their actions. What agents conjecture to be the consequences of these action
on the outcomes of the exchange institutions is crucial in determining their
optimal actions. The outcomes of this kind of models typically depend on
the particular conjectures the agents hold. For the models under discussion,
Hahn (1978) considers the analogon of rational expectations equilibrium, called
equilibrium in rational conjectures. Unfortunately, Gale (1978) found that re
quiring conjectural equilibrium to be rational in this sense seems to be asking
too much.
A more general approach to introducing exchange institutions in a general
equilibrium framework is taken in Vind (1983) and Grodal and Vind (1989).
Economies with exchange institutions are introduced in Vind (1983) as an ap
plication of social systems with coordinations.4 In such economies, consumers
2See, e.g., Krelle (1976) and Tirole (1988)
3For a readable introduction to this kind of topics see, e.g., Milgrom and Roberts (1992).
4The concept of equilibrium in social systems with coordination as introduced in Vind
(1983) generalizes social systems or abstract economies as introduced in Debreu (1952}. It
generalizes social systems in such a way that conjectures are incorporated explicitly within the
model, although its main purpose is to extend social systems with a concept of coordination.
