### Working paper

## Remarks on the Afriat's theorem and the Monge-Kantorovich problem

*The article presents the key aspects of successful investment process modeling for investment projects. A successful investment process means a series of investment decisions on timing, risk and investment objects and activities for their implementation, aimed to generating positive indicator "alpha", which is the maximum possible total return of specific investment project. In practice, there is no unique investment process, applicable to any investment decision, however, there are basic requirements that must be met to ensure a successful outcome of investment process.* *These requirements include availability of investment opportunities, forecasting skills and mechanism for investment project implementation.*

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.

We provide a review of methods for studying economic data and building economic indices based on the generalized nonparametric method for computing Konus-Divisia indices, with the focus on the case when the observed behavior is not consistent with the classical Pareto's model of a single rational representative consumer, but may be consistent with the generalized model with two or more rational representative consumers. Computing economic indices allows one to aggregate economic prices and consumption data from the level of individual goods to the level of some general categories of goods.

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.