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Advances in Mathematical Economics Volume 11

herausgegeben von: S. Kusuoka, A. Yamazaki

Verlag: Springer Japan

Buchreihe : Advances in Mathematical Economics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

A lot of economic problems can formulated as constrained optimizations and equilibration of their solutions. Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories. The series is designed to bring together those mathematicians who were seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking for effective mathematical tools for their researchers. Members of the editorial board of this series consists of following prominent economists and mathematicians: Managing Editors: S. Kusuoka (Univ. Tokyo), A. Yamazaki (Hitotsubashi Univ.) - Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier II), F. H. Clarke (Univ. Lyon I), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Fukuoka Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama National Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Hitotsubashi Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), K. Kamiya (Univ. Tokyo), K. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), T. Maruyama (Keio Univ.), H. Matano (Univ. Tokyo), K. Nishimura (Kyoto Univ.), M. K. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

Inhaltsverzeichnis

Frontmatter

Chapter 1. Optimal hedging strategies on asymmetric functions

Abstract.

We treat in this paper optimal hedging problems for contingent claims in an incomplete financial market, which problems are based on asymmetric functions. In summary, we consider the problem

$$\mathop{\rm min}\limits_{\vartheta\in\Theta} E[f(H - G_T(\vartheta))],$$

where H is a contingent claim, Θ, which is a suitable set of predictable processes, represents the collection of all admissible strategies, $G_T(\vartheta)$ is a portfolio value at the maturity T induced by an admissible strategy $\vartheta$ , and $f : \mathbf{R} \to \mathbf{R}_+$ is a differentiable strictly convex function with f(0) = 0. In particular, under the assumption that there exist two positive constants c ₀ and C ₁ such that, for any $x \in \mathbf{R}$ being far away from 0 sufficiently, $c_0|x|^p\leq f(x)$, and $|f^\prime(x)|\leq C_1|x|^{p-1}$, where 1 < p < ∞, we shall prove the unique existence of a solution and shall discuss its mathematical property.

Takuji Arai

Chapter 2. Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions

Abstract

Various notions of tightness for measurable multifunctions are introduced and compared. They are used to derive results on the existence of integrable selections for the sequential weak upper limit of a sequence of multifunctions. Similar questions are examined for multifunctions with values in a dual space. Some results are particularized in the single-valued case, and applications to the multidimensional Fatou Lemma, both in the primal and in the dual space, are derived. This is achieved under conditions weaker than or noncomparable to L ¹-boundedness.

Charles Castaing, Christian Hess, Mohamed Saadoune

Chapter 3. Core convergence in economies with bads

Abstract

We investigate how the presence of bads, causing disutility to consumers, affects the emergence of the price-taking behavior. Specifically, we give two examples of sequences of increasingly populous finite economies in which the core convergence property holds and, yet, for which there is a sequence of coalitions, one from each economy, such that the size of the coalition relative to the economy converges to zero but the share of the coalition in the aggregate consumption of bads converges to one. The limit atomless economy has a Walrasian equilibrium in one of the two examples but not in the other.

Chiaki Hara

Chapter 4. A distance and a binary relation related to income comparisons

Abstract

We define a distance and a binary relation among income distributions which is closely related to Lorenz dominance. An income distribution is represented by a vector (x ₁, x ₂,..., x _n) when the society under consideration consists of n individuals or households. The component x _i denotes the income of the ith individual and the sum $\sum_{i=1}^n x_i$ is the total wealth of the society. The distance is defined on the n-dimensional Euclidean space R ⁿ mathematically, and it gives indices of difference between two income distributions with not only the same total wealth but also the different total wealths. Thus, the distance might give a criterion for income distributions taking account of equity and efficiency.

Hidetoshi Komiya

Chapter 5. On preference relations that admit smooth utility functions

Abstract

We prove the existence of smooth utility functions for a class of preferences (closed preorders) on a subset X in ${\rm I}\!{\rm R}^n$ which satisfies $X=X+{\rm I}\!{\rm R}^n_+$. This class of preferences is given by the condition that adding one and the same positive vector to each of two comparable alternatives cannot affect the preference relation between them. Moreover, some its subclass consisting of total preferences admits linear utility functions. Also, we prove the existence of universal smooth utilities for preferences depending on a parameter. Our approach relies on our earlier results on continuous utilities for closed (non-total) preorders on metrizable spaces along with a particular device that enable to pass from a continuous utility to a smooth one.

Vladimir L. Levin

Chapter 6. Rational expectations can preclude trades

Abstract

We reconsider the no trade theorem in an exchange economy where the traders have non-partition information. By introducing a new concept, rationality of expectations, we show some versions of the theorem different from previous works, such as Geanakoplos (http://cowles.econ.yale.edu, 1989). We also reexamine a standard assumption of the no trade theorem: the common prior assumption.

Takashi Matsuhisa, Ryuichiro Ishikawa

Chapter 7. The Le Chatelier Principle in dynamic models of the firm

Abstract

This study examines the Le Chatelier Principle in intertemporal models of the firm with a delivery lag for capital. Adjustment costs are attached to labor and capital. Dynamic demands for labor and capital investment obey the principle when short-run and delivery-period factor price responses are compared. If own-adjustment parameters for quasi-fixed inputs are between zero and minus unity, a form of the principle holds when comparing delivery-period and steady-state factor price responses. Adding variable factors, the principle arises for quasi-fixed and variable factors in response to quasi-fixed factor prices but not to variable factor demands and variable factor input prices.

Robert J. Rossana

Chapter 8. Interdependent utility functions in an intergenerational context

Abstract

We investigate the question of representing nonpaternalistic functions (aggregators) in paternalistic form, which was posed by Ray (J. Econ. Theory 41:112–132, 1987), in an intergenerational setting. As in Hori (Jpn Econ. Rev. 52:137–155, 2001), the aggregators in this paper may differ across generations and depend possibly on the utility levels of all other generations. We discuss two approaches to deal with an infinite horizon. The first one explores monotonicity structures inherent in nonpaternalistic altruism. By means of lattice-theoretic arguments, we establish the existence of representations of nonpaternalistic functions in paternalistic form. The second approach uses the requirement of small degree of altruism.

Tomoichi Shinotsuka

Backmatter

Titel: Advances in Mathematical Economics Volume 11
herausgegeben von: S. Kusuoka
A. Yamazaki
Verlag: Springer Japan
Electronic ISBN: 978-4-431-77784-7
Print ISBN: 978-4-431-77783-0
DOI: https://doi.org/10.1007/978-4-431-77784-7

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Optimal hedging strategies on asymmetric functions

Chapter 2. Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions

Chapter 3. Core convergence in economies with bads

Chapter 4. A distance and a binary relation related to income comparisons

Chapter 5. On preference relations that admit smooth utility functions

Chapter 6. Rational expectations can preclude trades

Chapter 7. The Le Chatelier Principle in dynamic models of the firm

Chapter 8. Interdependent utility functions in an intergenerational context

Backmatter

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