最適化理論で重要な役割を果たしている Farkasの補題は凸集合の分離定数と深く関連しており、分離定数の議論にのせる前段階として有限錐の閉性が重要な役割を果たしていることがよく知られている。本講演では、この有限錐の閉性とで現れる Shapley-Folkmanの定理や Caratheodryの定理との関係を論ずる。

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The emergence of economics under Adam Smith rests on two methodological premises. First, there is a systematic understanding of the laws governing economic phenomena, achieved through the use of imaginative faculties, there is a method for recognizing these laws as interrelated within the conceptual realm. Second, there is a methodological approach to explain, through a few familiar principles, the reality of these laws through a few well-established principles, enabling their comprehension not only among scholars but also by the general public. The foundation of these methodological premises in Smith's approach is rooted in the method concerning laws that Sir Isaac Newton demonstrated in the Principia (1687) and Opticks (1704). In this paper, it is revealed that, with regard to Newton's method of discerning laws, Smith, in The History of Astronomy, embraced the idea that these laws could be recognized as a set of interconnected unifying principles within the sphere of imagination. Furthermore, in his Lectures on Rhetoric and Belles Lettres, Smith revealed that by employing a few such unifying principles in accordance with Newton's method, the laws could be explained as being interconnected on an imaginative level. It is also noted in this paper that the economic theory proposed in The Wealth of Nations employs this Newtonian method of natural philosophy through its moral–philosophical underpinning. Moreover, the accessible principles employed by Smith not only served to substantiate the reality of the so-called economic laws but also enabled the sharing of correct scientific knowledge, among both scholars and the general public from the emerging field of economics.

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Star refinement, a fundamental concept in homology theory is also important in mathematical economics, as in the proof of Begle [1]'s fixed point theorem used in Nikaido [5]. Although Tukey [7]'s proof has been referred to by many articles, it contains a serious error. For applied fields like mathematical economics, it would be useful and necessary to create a complete version of the proof. In this paper, we first clearly point out the above problem in Tukey [7], and then provide the entire proof.

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We investigate the impact of ambiguity on a competitive insurance market when companies are uncertain about the proportion of customers. Our study is motivated by the work of Zheng, Wang and Li [13], in which the authors explored feasible policies of a monopolist insurance provider facing a similar source of ambiguity. We demonstrate that insurance companies cannot establish any pooling equilibrium under the assumption of ambiguity aversion, whether wholly or partially ambiguous. Instead, any separating equilibrium must involve a fully insured policy for high-risk customers and an underinsured one for low-risk customers. Our findings affirm the robustness of the equilibrium presented in the original paper by Rothschild and Stiglitz [10], even in an ambiguous setting. Furthermore, employing a definition of epsilon contamination, we find that ambiguity significantly influences insurance companies' behavior. When the level of uncertainty is sufficiently high, a separating equilibrium always exists, as no insurer can offer any other pooling policy that benefits buyers. Therefore, a separating equilibrium is guaranteed.

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In this paper, we provide a Cobb-Douglas example of a dynamic formulation of market structure endogenization based on the notion of commodification technology in the overlapping generations model with money. The concept of commodification technology integrates two approaches of the commodity differentiation (like Lancaster [6] and Mas-Colell [7]) and the market viability (like Dubey et al. [4], and Bisin et al. [3]) to the formation of market structure, and is treated by the authors under the static general equilibrium framework (Urai et al. [12]).

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A general update rule for Choquet preferences is proposed and characterized. The update rule includes the major update rules such as the Dempster-Shafer rule (Shafer, 1976) and the Dempster-Fagin-Halpern rule (Dempster, 1967; Fagin and Halpern, 1991) as special cases. The axiomatic basis is established through a consistent counterfactual mapping that bridges between unconditional and conditional preferences. For any act, the mapping necessarily takes the form of a trinary act consisting of the minimum, maximum, and conditional certainty equivalent outcomes in the range of every act. The nature of the update rule is examined from the perspective of belief-by-belief updates.

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In this paper we consider the discrete-time model of Robinson, Solow and Srinivasan. We analyze this model without concavity assumptions on a utility function which represents the preferences of the planner and describe the structure of optimal programs on finite intervals.

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Poincaré-Miranda's theorem is an n-dimensional zero-point theorem, which was conjectured by Poincaré [6] and proved by Miranda [5]. Also, Miranda [5] proved its equivalence to Brouwer's fixed point theorem. The author [3] presented an n-dimensional intermediate value theorem by using Poincaré-Miranda's theorem. He applied it to two-person and three-person strategic games with two pure strategies, and showed a range of the payoffs that can be achieved by mixed strategies. In this paper, we extend the result of [3] to an n-person strategic game.

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In contrast to Mandler's (1999a; Theorem 6.1) impossibility result about the Sraffian indeterminacy of the steady-state equilibrium, we have proposed a simple example of overlapping generation economy in which generic indeterminacy occurs in the Sraffian steady-state equilibrium.

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Cournot's economics was deeply rooted in French mathematics in the nineteenth century. The main stream was Laplace-Lagrange's physico-mathematics. But in examining what drove Cournot to economics, we must not dismiss another stream, that is the tradition of social mathematics. It was probability theory which supported social mathematics as a principal tool of analysis. However, the mathematical machinery which Cournot used in economics was not probability theory but mathematical analysis while he succeeded the tradition of social mathematics. The transformation of mathematical methodology of economics due to Cournot was made possible by his peculiar grasp of the law of large numbers.

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This paper provides a rigorous and gap-free proof of the index theorem used in the theory of regular economy. In the index theorem that is the subject of this paper, the assumptions for the excess demand function are only several usual assumptions and continuous differentiability around any equilibrium price, and thus it has a form that is applicable to many economies. However, the textbooks on this theme contain only abbreviated proofs, and no known monograph contains a rigorous proof of this theorem. Hence, the purpose of this paper is to make this theorem available to more economists by constructing a readable proof.

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This paper studies how society should be risk-averse accordingly to how individuals are risk-averse. We consider two axioms, one that if everybody prefers one risky prospect over another so should the society, the other that if everybody is more risk-averse so should the society. We show that the two axioms imply dictatorship, in the sense that the social ranking over risky prospects has to be always identical with one individual's risk preference in each equivalence class of risk preferences which yield the same profile of ordinal preferences over deterministic outcomes.

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