Mathematical Economics Research Articles

Intuitive Mathematical Economics Series. Constrained Maximization and the Method of Lagrange Multipliers

A fundamental assumption in most of economic modeling is that people maximize their utility subject to a budget constraint. This, as well as many other economic problems, mathematically translate into problems of maximization with constraints. A powerful and widely used method to tackle some of these problems is the method of Lagrange multipliers. Yet, the exposition of such method in standard textbooks is rather formal and utilitarian. In this paper we try to present it emphasising the fundamental intuitions behind the method.

A PROGRESS REPORT ON MARXIAN ECONOMIC THEORY: ON THE CONTROVERSIES IN EXPLOITATION THEORY SINCE OKISHIO (1963)

AbstractThis report explores the development of exploitation theory in mathematical Marxian economics by reviewing the main controversies surrounding the proper definition of exploitation since the contribution of Okishio. The report first examines the debates on the Fundamental Marxian Theorem and Class‐Exploitation Correspondence Principle, developed mainly in the 1970s and 1980s, followed by the property relation theory of exploitation by Roemer. Then, the more recent exploitation theory proposed by Vrousalis and Wright is introduced. Finally, the report introduces and comments on recent axiomatic studies of exploitation by focusing on the work of Veneziani and Yoshihara.

Between mathematical formalism, normative choice rules, and the behavioural sciences: The emergence of rational choice theories in the late 1940s and early 1950s

This paper discusses why mathematical economists of the early Cold War period favored formal-axiomatic over behavioral choice theories. One reason was that formal-axiomatic theories allowed mathematical economists to improve the conceptual and theoretical foundations of economics and thereby to increase its scientific status. Furthermore, the separation between mathematical economics and other behavioral sciences was not as clear-cut as often argued. While economists did not modify their behavioral assumptions, some acknowledged the empirical shortcomings of their models. The paper reveals the multifaceted nature of rational choice theories reflected in the changing interpretations and roles of the theories in those early years.

Estimation of parameters of constant elasticity of substitution production functional model

Nonlinear model building has become an increasing important powerful tool in mathematical economics. In recent years the popularity of applications of nonlinear models has dramatically been rising up. Several researchers in econometrics are very often interested in the inferential aspects of nonlinear regression models [6]. The present research study gives a distinct method of estimation of more complicated and highly nonlinear model viz Constant Elasticity of Substitution (CES) production functional model. Henningen et.al [5] proposed three solutions to avoid serious problems when estimating CES functions in 2012 and they are i) removing discontinuities by using the limits of the CES function and its derivative. ii) Circumventing large rounding errors by local linear approximations iii) Handling ill-behaved objective functions by a multi-dimensional grid search. Joel Chongeh et.al [7] discussed the estimation of the impact of capital and labour inputs to the gris output agri-food products using constant elasticity of substitution production function in Tanzanian context. Pol Antras [8] presented new estimates of the elasticity of substitution between capital and labour using data from the private sector of the U.S. economy for the period 1948-1998.

Two-step values for games with two-level communication structure

TU games with two-level communication structure, in which a two-level communication structure relates fundamentally to the given coalition structure and consists of a communication graph on the collection of the a priori unions in the coalition structure, as well as a collection of communication graphs within each union, are considered. For such games we introduce two families of two-step values inspired by the two-step procedures staying behind the Owen value (Owen, in: Henn, Moeschlin (eds) Essays in mathematical economics and game theory, Springer, Berlin, pp 76–88, 1977) and the two-step Shapley value (Kamijo in Int Game Theory Rev 11:207–214, 2009) for games with coalition structure. Our approach is based on the unified treatment of several component efficient values for games with communication structure and it generates two-stage solution concepts that apply component efficient values for games with communication structure on both distribution levels. Comparable axiomatic characterizations are provided.

An Interview with William A. Barnett

An Interview with William A. Barnett

The Origin and Limitations of Modern Mathematical Economics: A Historical Approach

We have first demonstrated that Debreu’s view regarding the publication of The Theory of Games and Economic Behavior by von Neumann and Morgenstern in 1944 as the birth of modern mathematical economics is not convincing. In this paper, we have proposed the hypothesis that the coordinated research programs in the 1930’s, initiated by the Econometric Society and the Cowles Commission for Research in Economics with the objective of unifying economic theory, mathematics and statistics, can be regarded as the beginning of modern mathematical economics as well as econometrics. We have argued that this unification has failed to satisfactorily bridge the gap between mathematical economics and the real world economic issues. However, contrary to Marshall's view that mathematics is not an engine of inquiry in economics but is only a shorthand language, we have established in this paper that the application of mathematics in modern mathematical economics can, under certain conditions, produce economic results of value.

The Analysis of the Problem of Economic Mathematical Problems Reversed from the Ability of Logic Thinking in Students

This study aims to determine the level of logical thinking skills of solving ability on math problem of the economy, and the effect on the ability of logical thinking ability in solving mathematical economics at Mathematics Education students Muhammadiyah University Makassar. This research is an ex-post facto nature of causality with total population were 245 people from 8 different classes. Samples invoved 58 students consisting of 2 classes using a sampling technique . The research instrument consists of logical thinking ability test as many as 30 items and problem solving test ability by 5 items were valid and reliable before use. Data were analyzed using descriptive and inferential statistics (correlation and regression analysis). The research findings shows that: (1) The ability to think logically masiswa are in good enough category with an average value of 69.13; (2) The level of mathematical problem solving ability for masiswa economy is in the category quite well with the average value of 74.03; (3) The level of logical thinking ability masiswa correlated with mathematical problem solving ability with great economy of 94.5% relationship where that is located on a well once; and (5) the ability to think logically positive effect on the ability of solving problems with the effect of 89.1% while the remaining 10.9% is influenced by other variables outside the model. On conslusion, the ability to think logically positive effect on the ability of settlement capability, it can be concluded that the higher of logical thinking skills possessed by the student means the student problem-solving abilities are higher.

Formalising economics: social change, values, mechanics and mathematics in economic discourse

The mathematisation and formalisation of economic science has been one of the most important features of the development of economic science in the latter part of the twentieth century. What were the causes behind this ubiquitous mathematisation of economic science? Recent scholarship places excessive emphasis on the role and prestige of mathematics as a scientific tool. In this paper we argue this is inadequate, not least because it fails to account for the whole classical era when mathematical economists failed to have any impact. Explaining this failure is one of the main aims of this paper. Most accounts focus mostly on the intellectual factors involved failing to do justice to what amounts to a multifaceted and complicated phenomenon involving social, economic, political, intellectual, normative and institutional factors. Partly redressing the balance by bringing to the fore the various factors involved is the other prime aim of this paper.

Ricardian Inference: Charles S. Peirce, Economics, and Scientific Method

Standard histories of economics usually treat the “marginal revolution” of the midnineteenth century as both supplanting the “classical” economics of Smith and Ricardo and as advancing the idea of economics as a mathematical science. The marginalists – especially Jevons and Walras – viewed Cournot’s (1838) book on mathematical economics as a seminal work on which they could build. Surprisingly, the scientist, philosopher, and logician Charles S. Peirce discovered Cournot before the marginalist economists and possessed a deeper appreciation of his mathematical approach. While Peirce’s contributions to economics are limited, the influence of economics on his philosophy is subtle and not well understood. In a number of fragments, Peirce, who, despite Ricardo’s lack of mathematical form, nonetheless regarded him as a paradigmatic mathematical economist, refers to “Ricardian inference,” as a fundamental contribution to scientific method. Two, perhaps complementary, options are explored as to exactly what Peirce meant by Ricardian inference. On the one hand, he associates Ricardo with the “primipostnumeral syllogism,” which is a sort of generalization to uncountably infinite sets of what Peirce calls Fermatian inference (often referred to as mathematical induction). On the other hand, he holds up Ricardo as an exemplar of the “analytical method,” which is Peirce’s name for a hybrid form connecting analogy, abduction, and induction. On either account, economics plays a larger and more fundamental role in Peirce’s philosophy of science than is generally understood.

The Economics of Trade Liberalization: Charles S. Peirce and the Spanish Treaty of 1884

In the 1870s and 1880s, the scientist, logician, and pragmatist philosopher Charles S. Peirce possessed an advanced knowledge of mathematical economics, having mastered and criticized Cournot as early as 1871. In 1884 he engaged in a multi-round debate with the editors of The Nation over the economics of trade liberalization in the case of a proposed trade treaty with Spain concerning import tariffs on Cuban and Puerto Rican sugar. The debate is reconstructed and related carefully both to Peirce’s understanding of mathematical economics and to his philosophy of science.

Tropical convexity in Riesz spaces

Various properties of tropical convex sets of the positive cone of a Riesz space are established. Under suitable, but standard assumptions, we show that they have fixed points properties, for single valued and multivalued maps, analogous to those associated to the usual convex sets in topological vector spaces. Selection theorems and extension theorems similar to those of Michael and Dugundji also hold and, in the appropriate setting, tropical convex sets are absolute extensors for the class of metric spaces and absolute retracts. We also give some applications to standard problems of mathematical economics: Nash equilibria, existence of elements for non transitive preference relations and existence of equilibria for abstract economies in Riesz spaces under tropical convexity assumptions. This work can be conceived as a contribution to infinite dimensional idempotent analysis.

Ekonomista z Wilna - dorobek naukowy Władysława Mariana Zawadzkiego

Wladysław Marian Zawadzki (1885–1939), was born in Vilno, Polish economist, Minister of Treasure (1932–1935), pioneered mathematical economics in Poland (the Walrasian mathematical economist). The conservative professor of economics at the School of Commerce (SGH) in Warsaw and the Stefan Batory University of Wilno (1919–1931). The SGH had at this time the stronger team of theoreticians. It was led by Wladyslaw Zawadzki. He was a very thorough exposition of J.M. Keynes’s monetary analisis, contrasting it with his earlier ideas in the Treatise on Money and the Tract on Monetary Reform. Zawadzki regretted Keynes’s etatism but comforted himself and his readers that Keynes had changed his mind at least twice before The General Theory.

About Applications of the Fixed Point Theory

AbstractThe fixed point theory is essential to various theoretical and applied fields, such as variational and linear inequalities, the approximation theory, nonlinear analysis, integral and differential equations and inclusions, the dynamic systems theory, mathematics of fractals, mathematical economics (game theory, equilibrium problems, and optimisation problems) and mathematical modelling. This paper presents a few benchmarks regarding the applications of the fixed point theory. This paper also debates if the results of the fixed point theory can be applied to the mathematical modelling of quality.

Erratum to: Increasing quasiconcave co-radiant functions with applications in mathematical economics

Erratum to: Increasing quasiconcave co-radiant functions with applications in mathematical economics

Exploring South Africa’s southern frontier: A 20-year vision for polar research through the South African National Antarctic Programme

Mathematics is central to the modern discipline of economics, although there has always been some dispute about the value of this relationship. Leijonhufvud, in his satire of the economics profession, declared that the ‘Math-Econ’(mathematical economists) appeared to be the high priests of this tribe. There was debate on whether their work was ‘best to be regarded as religious, folklore and mythology, philosophical and ‘scientific’, or as sports and games’. In this commentary I want to consider whether the role of mathematics goes beyond ‘sports and games’ in economics and reflect on some of the ways in which mathematics is used in economics in general and in South African economics in particular.

Maximizing expected utility in the Arbitrage Pricing Model

We consider an infinite dimensional optimization problem motivated by mathematical economics. Within the celebrated “Arbitrage Pricing Model”, we use probabilistic and functional analytic techniques to show the existence of optimal strategies for investors who maximize their expected utility.

Optimizing staffing, quality, and cost in home healthcare nursing: theory synthesis.

To propose a new theory pinpointing the optimal nurse staffing threshold delivering the maximum quality of care relative to attendant costs in home health care. Little knowledge exists on the theoretical foundation addressing the inter-relationship among quality of care, nurse staffing, and cost. Theory synthesis. Cochrane Library, PubMed, CINAHL, EBSCOhost Web and Web of Science (25 February - 26 April 2013; 20 January - 22 March 2015). Most of the existing theories/models lacked the detail necessary to explain the relationship among quality of care, nurse staffing and cost. Two notable exceptions are: 'Production Function for Staffing and Quality in Nursing Homes,' which describes an S-shaped trajectory between quality of care and nurse staffing and 'Thirty-day Survival Isoquant and Estimated Costs According to the Nurse Staff Mix,' which depicts a positive quadric relationship between nurse staffing and cost according to quality of care. A synthesis of these theories led to an innovative multi-dimensional econometric theory helping to determine the maximum quality of care for patients while simultaneously delivering nurse staffing in the most cost-effective way. The theory-driven threshold, navigated by Mathematical Programming based on the Duality Theoremin Mathematical Economics, will help nurse executives defend sufficient nurse staffing with scientific justification to ensure optimal patient care; help stakeholders set an evidence-based reasonable economical goal; and facilitate patient-centred decision-making in choosing the institution which delivers the best quality of care. A new theory to determine the optimum nurse staffing maximizing quality of care relative to cost was proposed.

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.

The cultures of mathematical economics in the postwar Soviet Union: More than a method, less than a discipline

The cultures of mathematical economics in the postwar Soviet Union: More than a method, less than a discipline