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## Über dieses Buch

The papers collected in this volume are contributions to T.I.Tech./K.E.S. Conference on Nonlinear and Convex Analysis in Economic Theory, which was held at Keio University, July 2-4, 1993. The conference was organized by Tokyo Institute of Technology (T. I. Tech.) and the Keio Economic Society (K. E. S.) , and supported by Nihon Keizai Shimbun Inc .. A lot of economic problems can be formulated as constrained optimiza­ tions and equilibrations of their solutions. Nonlinear-convex analysis has been supplying economists with indispensable mathematical machineries for these problems arising in economic theory. Conversely, mathematicians working in this discipline of analysis have been stimulated by various mathematical difficulties raised by economic the­ ories. Although our special emphasis was laid upon "nonlinearity" and "con­ vexity" in relation with economic theories, we also incorporated stochastic aspects of financial economics in our project taking account of the remark­ able rapid growth of this discipline during the last decade. The conference was designed to bring together those mathematicians who were seriously interested in getting new challenging stimuli from economic theories with those economists who were seeking for effective mathematical weapons for their researches. Thirty invited talks (six of them were plenary talks) given at the conf- ence were roughly classified under the following six headings : 1) Nonlinear Dynamical Systems and Business Fluctuations, . 2) Fixed Point Theory, 3) Convex Analysis and Optimization, 4) Eigenvalue of Positive Operators, 5) Stochastic Analysis and Financial Market, 6) General Equilibrium Analysis.

## Inhaltsverzeichnis

### The Bargaining Set in Large Finite NTU Exchange Economies

Abstract
The bargaining set was originally defined by Aumann and Maschler [6]. Several different definitions have been subsequently proposed; the most frequently used definition was proposed by Davis and Maschler [8]. In the exchange economy context we consider here, the core consists of all allocations such that no coalition can propose an alternative set of trades which is feasible for the coalition on its own and which makes all of its members better off. All definitions of the bargaining set restrict the ability of coalitions to block (“object to”) an allocation, by taking into account the possibility that a second coalition might propose yet another set of trades (“counterobject”) and thereby cause some members to defect from the first coalition. In the Aumann-Maschler and Davis-Maschler definitions, the original objection is proposed by a single individual known as the leader of the objection; any counterobjecting coalition must exclude this leader. An objection is said to be justified if there is no counterobjection; the bargaining set consists of all allocations with no justified objection.
Robert M. Anderson

### Epi-convergence of Integral Functionals Defined on the Space of Measures

Abstract
This paper is concerned with the epi-convergence of integral functionals defined on the space of vector measures with bounded variation with values in a reflexive separable Banach space.
Charles Castaing, Vincent Jalby

### Multiple—Phase Economic Dynamics

Abstract
We observe in economics, as in other fields, that quite different forces or relationships govern behavior in differing situations of state. Multiple—phase dynamic models formalize this fact.
Richard H. Day

### Stabilizing Unstable Systems

Abstract
A nonlinear difference equation which diverges cyclically may be stabilized when time-lag is introduced in a certain way. It is shown by using a numerical example that while stabilization is impossible when the resulting difference equation is of order two, it can be made stable when the order is not less than three. One application is made to stabilize the examples of global instability of competitive equilibrium by Scarf. Some suggestions are made concerning the computation of a fixed point, and the shrinkage of chaotic areas.
Takao Fujimoto

### On First Order Sufficient Conditions for Constrained Optima

Abstract
Some remarks are made on the results obtained by A. Mukherji[15] for mathematical programming problems involving quasi-concave functions and some extensions of the same results are established. Moreover, for the same problems and under no generalized concavity assumptions, some first order local optimality conditions are obtained.
Giorgio Giorgi

### Expectations Driven Nonlinear Business Cycles

Abstract
There are two traditional conflicting views about the workings of a market economy. The socalled “Classical” school stresses the virtues of free markets and their intrinsic internal stability. According to that school, persistent, nonexplosive fluctuations should be essentially due to repeated external macroeconomic shocks to the “fundamental” characteristics of the system (technology, tastes, resources). Models of the “Classical” vintage typically assume that expectations are self-fulfilling, i.e. every individual’s assessment of the future (a probability distribution) is correct at any moment given his information. According to that view, expectations cannot be an independent source of economic fluctuations. There is another view, often associated to so-called “Keynesian” thinking, that starts with the observation that individual prophecies about aggregate outcomes often tend to be self-fulfilling if shared by sufficiently many people, and that suggests this phenomenon as a potential source of significant internal instability of socioeconomic systems in the absence of any outside regulation. According to that view, aggregate changes of expectations are likely to occur rather frequently in an unpredictable way and are potentially major sources of endogenous business fluctuations (market psychology, animal spirits).
Jean-Michel Grandmont

### Cores, Almost Competitive Prices, and the Approximate Optimality of Walrasian Allocations in Discrete Spaces

Abstract
This paper is devoted to problems arising when all or some goods are indivisible. Part One deals with the properties of the core in such situations, Part Two with the optimality properties of the Walrasian (competitive) allocations. The purpose of Part One is to show how certain relationships between the core and prices can be extended (with modifications) to exchange economies in which all or some commodities are indivisible. The proposition3 stating that a Walrasian allocation is in the core (provided preferences are selfish) remains valid without any additional assumptions when all goods are indivisible, as can be seen by inspecting the standard proofs. (see, e.g., Debreu and Scarf, Th-m 1, p.240, or Arrow and Hahn, Ch. 8, Th-m 1, p.187). [This is largely due to the fact that the blocking definition requires that strict preference hold for each member of a blocking coalition. Hence, in particular, a core allocation is only required to be Weakly Pareto Optimal (i.e., Pareto Efficient in the sense of Arrow and Hahn or Varian) rather than Pareto Optimal as defined, e.g., in Koopmans, p.46.4 If a ‘strict core’ were defined as the set of allocations ‘strongly unblocked’ in the sense of Def. 6, p.196, in Arrow and Hahn, then additional assumptions, such as local non-satiation, would be needed to guarantee that a Walrasian allocation belongs to the strict core.]
Leonid Hurwicz

### Cooperative Processing of Information

Abstract
The Nash equilibrium concept of a normal-form game (Nash [15]) has been extended in several directions in the past: One direction is to introduce asymmetric information explicitly into the normal-form game. The resulting model is Harsanyi’s [9] Bayesian game. His Bayesian equilibrium concept extends the Nash equilibrium concept to this new framework.
Tatsuro Ichiishi

### A Survey of Stochastic Differential Equations

Abstract
Let us consider a system, dynamical,biological or economical, that is determined by a finite number of parameters:
$$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{x}=({{x}^{1}},{{x}^{2}},\ldots,{{x}^{r}})\in{{R}^{r}}$$
Kiyosi Itô

### Stability and Oscillations in a Dynamical Kaldorian Model

Abstract
In a classic paper N.Kaldor (1940) presented a model of a closed economy and demonstrated the existence of a periodic orbit which may be interpreted as a trade cycle in an autonomous system.1 In establishing this result he assumed not only that investment and saving decisions are made without any time lag, but also that investment is adjusted to saving instantaneously so far as the adjustment is possible by a small change in output. Thus, in his model, non-negligible adjustment period is taken into account only in installing capital and making it ready for production.
Kunio Kawamata

### On Large Games with Finite Actions: A Synthetic Treatment

Abstract
In this expository paper, we untangle the relationship between anonymous and non-anonymous versions of the theory of large games. Our treatment shows that the two formulations collapse to one essentially equivalent theory in the case of finite action spaces, and exhibit rich differences only when this finiteness is dispensed with.
M. Ali Khan, Yeneng Sun

### On a Method Constructing Morse Flows

Abstract
By enumerating a treatment of a variational problem of harmonic type in illustration, we shall propose a method constructing Morse flows of variational problems.
Norio Kikuchi

### On Covering Theorems of a Simplex and Their Generalizations

Abstract
Since Scarf established non-emptiness of the core of a balanced nontransferable utility game in 1967, others have attempted to prove it in a simple way. Most of them reduced the problem to that of a covering of a simplex in the Euclidean space. Thus, we have obtained several covering theorems of a simplex in the course of the endeavor to prove the Scarf’s theorem. We shall give a theorem unifying these covering theorems.
Hidetoshi Komiya

### Positive Nonlinear Systems in Economics

Abstract
In many cases an economic system can be modelled by a mapping transforming a state of the economic system at a certain period of time into the state of the system at the next period. If the transformation under consideration can be assumed to be linear then the well-established theory of linear operators can be applied; thereby spectral theory, including Perron-Frobenius theory for positive matrices and positive linear operators, is of particular importance. Very often, however, linearity is not an appropriate idealization, in which case a rigorous analysis may become very difficult or even impossible. It is this state of affairs which brings positive nonlinear systems into play, this not only in economics. Positivity and related mathematical properties are quite natural assumptions in economics. The state space is often given, e.g., if states are described by quantities or prices, by the positive orthant (or some more general convex cone) in Euclidean space. The transformation of such a state space may possess additional properties related to positivity as various forms of monotonicity. This is the case for the two economic problems considered in this paper: Balanced growth in a nonlinear multisectoral framework and price setting among several production units which depend on each other by technology. Given the transformation T mapping the state space K, a convex cone, into itself, the following questions will be addressed: Does there exist a unique equilibrium, that is does the fixed point equation Tx = x possess a unique solution x є K (up to a positive scalar)?
Ulrich Krause

### Option Replication Cost with Transaction Costs

Abstract
Recently there appeared several papers which discussed the option replication in the case that transaction costs exist (e.g. Leland(1985), Merton(1990), Boyle-Vorst(1992)). In the present paper, we introduce the notion of option replication cost and see the effect of the existence of the transaction costs.
Shigeo Kusuoka

### Fixed Point and Finite Dimensional Invariant Subspace Properties for Semigroups and Amenability

Abstract
A well-known theorem of Markov-Kakutani [5, p. 456] asserts that if S is a commutative semigroup, then S has the following fixed point property: (1) whenever S = {T S ; sS} is a representation of S as affine continuous mappings from a non-empty compact convex subset K of a separated locally covex space (i.e. T s x + (1 − λ)y) = λT s (x) + (1 − λ)T s (y), 0 ≤ λ ≤ 1, x, yK), then K contains a common fixed point for S.
Anthony To-Ming Lau

### Pricing of Bonds and their Derivatives with Multi-factor Stochastic Interest Rates: A Note

Abstract and Introduction
Vasicek(1977) derived a Bond Pricing formula for the interest rate following a single factor Ornstein-Uhlenbeck process. Along with this line, Jamshidian(1989) derived a bond option pricing formula, and Chen (1992) derived formulae for futures and futures option on bonds.
Miura Ryozo, Kishino Hirohisa

### Non-Linearity and Business Cycles in a Two-Sector Equilibrium Model: An Example with Cobb-Douglas Production Functions

Abstract
This study presents a two-sector optimal growth model with Cobb-Douglas production functions in which optimal dynamics exhibits sharp non-linearity giving rise to cyclical optimal paths. This result demonstrates that such optimal paths may appear for any value of the discount factor of future utilities. Moreover, once a cyclical optimal path appears for a particular value, it appears for any value that discounts the future utilities stronger than that particular value.
Kazuo Nishimura, Makoto Yano

### Methods of Duals in Nonlinear Analysis

Lipschitz Duals of Banach Spaces and Some Applications
Abstract
In the functional analysis for linear operators, there are many types of useful methods, techniques and tools. On the other hand, for nonlinear operators, it seems that functional analysis does not have so many fundamental methods, though various types of fixed point theorems have been investigated. Such a situation it made us eager to find other applicable methods to the nonlinear analysis.
Ikuko Sawashima

### Swimming below Icebergs

Abstract
You are swimming close to an iceberg in the ocean. You calculate at what slope you have to swim down so that, whatever the direction in which you swim, you can be sure that you will not collide with the iceberg. We shall see that, provided that the lower surface of the iceberg is convex, this limiting slope is intimately related to the existence of subtangents to the iceberg that satisfy varions conditions. These considerations lead to generalizations of Rockafellar’s Maximal Monotonicity Theorem, and also of recent results on the existence of subtangents separating the epigraphs of proper convex lower semicontinuous functions from nonempty bounded closed convex sets.
S. Simons

### Comparative Statics and Algorithms for Finding Economic Equilibria

Abstract
Consider an excess demand function Z : ℝ + l → ℝ l , pZ(p), where ℝ + l is the set of price vectors p = (p l,..., p l), p i ≥ 0 and the value of Z are taken in commodity space ℝ l . For example, Z = DS, demand less supply, and D, S are derived from a microeconomical setting. This is the approach in [Smale], where some background for this note may be found.
Steve Smale

### Classification of chaos and a large deviation theory for compact dynamical systems

Abstract
A survey will be given on the large deviation theory aspect of the classification of chaos with a slight generalization of previous results.
Yoichiro Takahashi

### On a Perturbation of Dynamic Programming

Abstract
Recently, in view of practical problems, dynamic control problems have been investigated by many authors. In general, a mathematical formulation for these problems is given and, in this formulation, an optimal value and an optimal policy are mainly studied. Then, under some conditions, it will be necessary to find out an optimal value and an optimal policy. But, in order to show the existence of an optimal policy, we need to assume the stronger condition such that the control space is compact. So, when the compactness condition is excluded, the ε—optimal policies will be mainly studied.
Kensuke Tanaka

### Bargaining Sets in Continuum Economies

Abstract
Aumann and Maschler [3] originally introduced the concept of the bargaining set. The bargaining set differs from the core in that it takes account of the reaction of others when a coalition engages on an improving move as an objection to a proposed allocation. Mas-Colell [9] proposed a redefinition of the original concept so that it does not depend upon distinguished individual agents and becomes well defined in continuum economies.
Akira Yamazaki

### Backmatter

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