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## Curvature and the elasticity of substitution: What is the link?

Relation between curvature and the elasticity of substitution is the old question important for economic theory. Opinions of economists concerning presence or absence of a link between these two concepts radically diverge. Also now there is a steady trend of the use of the Arrow-Pratt coefficient of relative risk aversion and the coefficient of relative prudence as characteristics of utility functions and production functions even in non-stochastic models, and these two coefficients are also commonly interpreted as measures of curvature. The purpose of the paper is to contribute to clarification of the links between all these concepts. We suggest a simple unifying approach based on the notions of prototype functions and osculating curves. In framework of this approach we easily derive the classic geometric curvature and show the relations between the Arrow-Pratt coefficient, the prudence coefficient, the elasticity and the elasticity of substitution. As an example, demonstrating the role of such relations in economic models, we study a simple macroeconomic model with a non-homothetic production function.

Properties of increasing positively homogeneous functions are studied; in particular, their representations by use of tropical inner products with coefficients chosen from tropical support sets are described. An application to a model of economic complementarityand weak links is developed. It is shown that weak links do not necessary bound total factor productivity from below but in some cases constraint it from above.

In the present paper the game theory is applied to an important open question in economics: providing microfoundations for often-used types of production function. Simple differential games of bargaining are proposed to model a behavior of workers and capital-owners in processes of formation of a set of admissible factor prices or participants’ weights (moral-ethical assessments). These games result, correspondingly, in a factor price curve and a weight curve – structures dual to production function. Ultimately, under constant bargaining powers of the participants, the Cobb-Douglas production function is received.

The monograph presents the results of calculations of the human capital dynamics and structure for the Russian economy in 1991-2012 years, using the method of accumulated costs by analogy with the calculation of the fixed capital volume. Analysis of the human capital contribution implemented on the basis of the production function model; original mathematical-statistical methods providing stability and economic interpretability of the results developed.

One of the sections of economic and mathematical modeling is a section dedicated to the modeling of economic dynamics of large systems. The model in this section is a system of equations and inequalities, describing a closed production cycle - from the formation of the main production resources to replenish their next production cycle due to the distribution of results for the resumption of production and growth of resources. Such a closed system of equations and inequalities reflects the fundamental relationship of real economic production systems, and therefore can be used in a variety of economic experiments. With their help it is possible to determine the results of the regulatory impact on the economy, to assess the significance of various measures of state regulation - from changes in the tax system to a variety of protectionist measures. This book is a new model of economic dynamics, based on the principles of complex-economy. Using models of complex variables allows us to describe these economic processes and relationships that are either difficult or impossible to describe using models of real variables.

We discuss how the curvature and the strain density of an atomic lattice generate the quantization of graphene sheets as well as the dynamics of geometric quasiparticles propagating along the constant curvature/strain levels. The internal kinetic momentum of a Riemannian oriented surface (a vector field preserving the Gaussian curvature and the area) is determined.

The Tait-Kneser theorem states that the osculating circles of a plane curve with monotonic curvature are pairwise disjoint and nested. We discuss this theorem and a number of its variations.

The new economic-mathematical model based on complex variables theory and the new approach to complex variables usage in economics are suggested in the article. The comparison of modeling results of actual production processes using Cobb-Douglass production function and complex variables production function is conducted. It is shown that the instrumental base of economicmathematical methods can be widen with usage of complex variables theory.

Selected works of George Kleiner on economics and mathematics in occasion of his 70th birthday.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.