Mathematical Economics Research Articles

On complete classes of valuated matroids

We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We exhibit a family of valuated matroids that are not R-minor based on sparse paving matroids. Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials: it reveals the limitations of known construction operations.

Against formalization as translation and the prospects for mathematical Austrian economics

Against formalization as translation and the prospects for mathematical Austrian economics

Mathematical Economics in the Era of Socialism and the Transition to the Market (Part Two)

Mathematical Economics in the Era of Socialism and the Transition to the Market (Part Two)

Mathematical Economics in the Era of Socialism and the Transition to the Market (Part One)

Mathematical Economics in the Era of Socialism and the Transition to the Market (Part One)

Critical Analysis of Increasingly Mathematical Economics in the Modern Society

Critical Analysis of Increasingly Mathematical Economics in the Modern Society

A Study on the Institutional Context and Dynamics Surrounding the Emergence of SARS-CoV-2

This dissertation presents an in-depth analysis of the institutional context and dynamics surrounding the emergence of SARSCoV-2, the virus responsible for the COVID-19 pandemic. Focusing on the case of Brazil, the study employs a multi-disciplinary approach that integrates mathematical economics, epidemiology, and public health policy analysis. Key aspects include examining the initial outbreak in Wuhan, the virus's rapid global spread, and the specific impacts and responses in Brazil. The research employs mathematical models to understand the dynamics of virus transmission, the effectiveness of various public health interventions, and the socio-economic consequences of the pandemic. Special attention is given to Brazil's vaccination campaigns and their role in pandemic control. The findings offer valuable insights into the complexities of managing a public health crisis in a socio-economically diverse country like Brazil and underscore the importance of timely and coordinated responses to global health emergencies.

New Type of Extended Soft Set Operation: Complementary Extended Union Operation

Soft set theory was proposed by Molodtsov in 1999 to model some problems involving uncertainty. It has a wide range of theoretical and practical applications. Soft set operations constitute the basic building blocks of soft set theory. Many kinds of soft set operations have been described and applied in various ways since the inception of the theory. In this paper, to contribute to the theory, a new soft set operation, called complementary extended union operation, is defined, its properties are discussed in detail to obtain the relationship of each operation with other soft set operations, and the distributions of these operations over other soft set operations are examined. We obtain that the complementary extended union operation along with other certain types of soft set operations construct some well-known algebraic structure such as Boolean Algebra, De Morgan Algebra, semiring, and hemiring in the set of soft sets with a fixed parameter set. Since Boolean Algebra is fundamental in digital logic design, computer science, information retrieval, set theory and probability; De Morgan Algebra in logic and set theory, computer science, artificial intelligence, circuit design; semirings in theoretical computer science, optimization problems, economics, cryptography and coding theory, and hemirings in combinatorics, mathematical economics, theoretical computer science, these algebraic structures provide essential tools for various applications, facilitating the analysis, design, and optimization of systems across many disciplines, and thus this study is expected to contribute to decision-making methods and cryptography based on soft sets.

A general law of the iterated logarithm for non-additive probabilities

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-additive probabilities have been used to describe the phenomena which are generally non-additive. In this paper, we further study the law of the iterated logarithm (LIL) for non-additive probabilities, based on existing results. Under the framework of sublinear expectation initiated by Peng, we give two convergence results of Vn≔∑i=1nXinϕ(n) under some reasonable assumptions, where {Xi}i=1∞ is a sequence of random variables and ϕ is a positive nondecreasing function. From these, a general LIL for non-additive probabilities is proved for negatively dependent and non-identically distributed random variables. It turns out that our result is a natural extension of the Kolmogorov LIL and the Hartman–Wintner LIL. Theorem 1 and Theorem 2 in this paper can be seen an extension of Theorem 1 in Chen and Hu (2014).

Recent advances on testability in economic equilibrium models

The revealed preference program started by Paul Samuelson brought a Popperian view of what constituted true scientific discovery to economic theory.11Samuelson (1938). In his Foundations, Samuelson states that “meaningful theorems” are hypotheses “about empirical data which could conceivably be refuted”.22See Samuelson (1947), p. 4. He worried that by appealing to unobservable fundamentals such as preferences, the body of economic theory built upon the premise of rational individual choice lacked a scientific foundation.While the program quickly succeeded in providing an empirical counterpart for the individual choice problem, its application to social outcomes, namely market interactions and games, proved more difficult, and only in the 1990s did the profession start to develop the testable implications of equilibrium concepts in these contexts. Carvajal et al. (2004) covered the early literature that followed the seminal contributions of Brown and Matzkin (1996) and Sprumont (2000). This survey is an update on the state of that literature covering the results of the last two decades. As the reader will see, the Journal of Mathematical Economics has played a key role in developing these ideas.

The Frame for the Not-Yet Existent: How American, European, and Soviet Scholars Jointly Shaped Modern Mathematical Economics

This article tells the story of the first international topological conference in Moscow (1935), an outstanding event that, for the first time, brought together the most notable American, European, and Soviet mathematicians, including those who would later play decisive roles in the mathematization of economics: John von Neumann, Leonid Kantorovich, and Albert W. Tucker. The fact that Kantorovich was in contact with von Neumann and his closest colleagues, Solomon Lefschetz and Garrett Birkhoff, is hardly appreciated in the histories of mathematics and mathematical economics. Their brief academic exchange was interrupted by the increasing international isolation of Soviet mathematics and by the wars that ensued. The article provides a historical account of the conference and traces the intellectual and personal affinities of Soviet and non-Soviet mathematicians, as well as their conceptual innovations. It argues that the conference, as a singular event linking several research communities, mattered for the development of various formal frameworks and their dissemination, contributing to the intellectual landscape in which postwar mathematical economics could emerge. The article calls for a deeper analysis of conceptual affinities and motivations in applying mathematics to economics and for a more nuanced narrative linking these motivations to social and political contexts of economic modeling.

Applying Pure Mathematics: IMPA and the Entanglements of Mathematical Economics in Brazil

This article studies the entanglements of mathematical economics in Brazil by delving into the emergence of a scientific community identified with the subject at the Instituto de Matemática Pura e Aplicada (IMPA). It traces how networks built since IMPA's early years in the 1950s made economics a viable alternative to applied mathematics in an institute mostly dedicated to research in pure mathematics. The article examines how interests in mathematical economics remained active during IMPA's expansion beginning in the late 1960s. This process culminated with the creation of the mathematical economics graduate program at IMPA in 1980 by Aloisio Araujo. The article follows how the program could build strong connections to the Escola de Pós-Graduação em Economia (EPGE), an important department of economics in Brazil, helping to shape recent developments in the center. The article thus aims to reveal how historical connections and entwinements of economics and mathematics groomed at IMPA have affected recent developments in the Brazilian economics community.

Recent advances on uniqueness of competitive equilibrium

This article reviews the recent advances in the uniqueness and multiplicity of competitive equilibria in models arising in mathematical economics, finance, macroeconomics, and trade.

Local Entanglements in the History of Mathematical Economics

This essay introduces a special issue of History of Political Economy titled “Local Entanglements in the History of Mathematical Economics.” The articles in the special issue share a particular historiographical approach, characterized by the name of “local entanglements.” The introduction defines the approach and explains why it can contribute in an original way to the history of economics.

Staatswissenschaften and the Mathematical Policy Science of Jan Tinbergen

The rise of mathematical economics is typically understood as a fundamental shift in the language and technique of economic theorizing. This article argues that an examination of Tinbergen's work demonstrates that a similar “mathematical” turn occurred in economic policy. This article contextualizes Tinbergen's professional career at the Dutch Central Bureau of Statistics and his early policy-oriented contributions as exemplified by the applied policy reports that he wrote between 1931 and 1936. His econometric work, including his seminal 1936 model of the Dutch economy and the closely related League of Nations model of the US economy, was rooted in his vision that business cycle research functioned in the service of the state. His work was aimed not at uncovering fundamental or structural relationships in the economy but at practical interventions in the economy by a state that developed control over new instruments of macroeconomic management. As such it can be and has been analyzed as an important contribution to economics as an engineering science. But this article demonstrates that Tinbergen also explicitly depicted the economy from the perspective of the policymaker and that he was attentive to the changing institutional order of the national and international economy. Tinbergen's work in this respect owed a great deal to the more political-economic tradition of German Staatswissenschaften than is typically acknowledged.

Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

Pareto’s legacy in the Italian tradition: the case of mathematical economics

Pareto’s personality had a great influence on the evolution of economic theory in the last century at least in the Italian context. At the beginning of the twentieth century a small but combative circle of followers aimed to defend and spread the new Paretian approach based on the idea of general economic equilibrium. Paretian scholars were well recognizable in the Italian context for their contributions to mathematical and statistical aspects. The purpose of this contribution is to outline the essential features of the Italian Paretian school in the field of pure economics or as we would say today of mathematical economics. This favorable position of the Italian tradition in the field of mathematical economics was basically due to two circumstances. The first was represented by the legacy of Pareto and his school while the second was to be found in the fact that among the students of Pareto there was a pure mathematician of great value Luigi Amoroso. We see then that an important voice even if poorly valued was that of Bruno de Finetti who published a remarkable series of articles on mathematical economics in the second half of the 1930s. After World War II the Paretian tradition in this field weakened considerably and effectively disappeared internationally.

Fifty years of mathematical growth theory: Classical topics and new trends

We present an overview of selected contributions of the Journal of Mathematical Economics’ authors to growth theory in the last half century. We start with the classical optimal growth theory within a benchmark multisector model and outline the successive developments in the analysis of this model, including the turnpike theory. Different refinements of the benchmark are considered along the way. We then survey the abundant literature on endogenous fluctuations in two-sector models. We conclude with two strong trends in the recent growth literature: green growth and infinite-dimensional growth models.

Housing markets since Shapley and Scarf

Shapley and Scarf (1974) appeared in the first issue of the Journal of Mathematical Economics, and is one of the journal’s most impactful publications. As we approach the remarkable milestone of the journal’s 50th anniversary (1974–2024), this article serves as a commemorative exploration of Shapley and Scarf (1974) and the extensive body of literature that follows it.

On equilibria in constrained generalized games with the weak continuous inclusion property

In this paper, we present results that use Himmelberg's fixed point theorem to highlight substantive trade-offs between compactness, continuity and convexity postulates in the setting of a constrained generalized game. The primary contribution is a focus on weakening the compactness assumption on the action sets and on two versions of the continuity assumption encapsulated as the continuous inclusion property (CIP). We show that the results are invariant to cardinality assumptions on the set of agents and on the dimension of the space in which the action sets are situated. This yields as special cases recent results in both mathematical economics and applied mathematics, and underscores the development of a rich and useful theory in bringing these two registers and two communities together.

Classification of quasi‐product production models with minimal isoquants

The study of the minimality of (hyper)surfaces is a fundamental problem in mathematics with major applications in both fundamental and applied sciences. Recently, the minimality of production models in microeconomics has been investigated by several authors who obtained important classification results for quasi‐sum and quasi‐product production functions with two and three inputs. In the present work, we deal with the minimality of isoquants of production functions with arbitrary number of inputs, this being a key concept used in making the best decision for optimizing the production costs. In the main result of the article, we give the complete classification of quasi‐product production functions whose isoquants are minimal, showing that there are exactly nine production models with such a property. Among them, we can identify a particular classic model already widely used in economic analysis (viz., a Cobb–Douglas model with unit output elasticities with respect to some production factors) and various production models completely unknown until now and which will open new opportunities for applied research in the field of mathematical economics.