A. A. Shananin

Study of the Bellman equation in a production model with unstable demand

A production model with allowance for a working capital deficit and a restricted maximum possible sales volume is proposed and analyzed. The study is motivated by the urgency of analyzing well-known problems of functioning low competitive macroeconomic structures. The original formulation of the task represents an infinite-horizon optimal control problem. As a result, the model is formalized in the form of a Bellman equation. It is proved that the corresponding Bellman operator is a contraction and has a unique fixed point in the chosen class of functions. A closed-form solution of the Bellman equation is found using the method of steps. The influence of the credit interest rate on the firm market value assessment is analyzed by applying the developed model.

Production model in the conditions of unstable demand taking into account the influence of trading infrastructure: Ergodicity and its application

A production model with allowance for a working capital deficit and a restricted maximum possible sales volume is proposed and analyzed. The study is motivated by an attempt to analyze the problems of functioning of low competitive macroeconomic structures. The model is formalized in the form of a Bellman equation, for which a closed-form solution is found. The stochastic process of product stock variations is proved to be ergodic and its final probability distribution is found. Expressions for the average production load and the average product stock are found by analyzing the stochastic process. A system of model equations relating the model variables to official statistical parameters is derived. The model is identified using data from the Fiat and KAMAZ companies. The influence of the credit interest rate on the firm market value assessment and the production load level are analyzed using comparative statics methods.

Estimation of the rate of return generated by a pool of investment projects in the continuous-time model of optimal investment

This work is devoted to the estimation of the rate of return generated by a pool of renewable investment projects. Cantor-Lippman’s formulation of the model for the case of continuous time is proposed. The result is substantiated, making it possible to classify pools of investment projects into arbitration, ineffective, or standard ones. For each class the rate of return for the pool of investment projects is found. The pools’ classification and the calculation of their rate of return are based on the functions of the upper envelope of the Laplace transform of the payment flow functions of the investment projects. It is shown that for the case of a standard pool the rate of return can be obtained by calculating the minimal positive root of the function of the upper envelope. An example is analyzed illustrating how the interaction between the developed and developing economies can be analyzed in terms of investment projects.

On the Cauchy–Gelfand Problem

The problem posed by Gelfand on the asymptotic behavior (in time) of solutions to the Cauchy problem for a first-order quasilinear equation with Riemann-type initial conditions is considered. By applying the vanishing viscosity method with uniform estimates, exact asymptotic expansions in the Cauchy–Gelfand problem are obtained without a priori assuming the monotonicity of the initial data, and the initial-data parameters responsible for the localization of shock waves are described.

Idempotent analogs of theorems on nonnegative matrices and their applications to the analysis of economic information

Problems arising in the analysis of economic data using idempotent analogs of theorems on nonnegative matrices are considered. Examples of such problems arise in the analysis of arbitrage chains in currency markets, in the calculation of economic indexes, and in the Kantorovich-Makarov model. It is shown that algorithms based on idempotent analogs of theorems on nonnegative matrices make it possible to develop a nonparametric method for the analysis of stock and currency markets.

The Chebyshev-Markov-Krein algorithm for analyzing microphysical processes in hail clouds

A modified Chebyshev-Markov-Krein algorithm is proposed for controlling processes governed by kinetic equations.

Variation Principles in Models of Economic Equilibrium

There are two approaches to modeling the price structure in mathematical economics. On the one hand, the Lagrange multipliers are conventionally interpreted as prices in optimization problems, in which the constraints represent balance relations describing the production capacity of the economic system. In case of linear optimization models of interindustry balance, prices are determined by the solution of a dual linear programming problem. On the other hand, prices in economic equilibrium models are determined from the equilibrium between supply and demand. We suggest variation principles for economic equilibrium models and a scheme for the construction of dual problems to determine equilibrium prices. These principles are formulated in the form of variational inequalities (Aubin 1983). If the total consumer demand functions satisfy the Probenius integrability conditions and obey the Hicks law, then the variation principles degenerate into an ordinary pair of dual optimization problems corresponding to the first approach to modeling the price structure.

A Generalization of the Nonparametric Method in Case of Trade Statistics not Satisfying the Hypothesis of Rational Behavior

A generalization of the nonparametric method for constructing demand and price indices in case of nonrational consumer behavior is investigated. We consider relationships between characteristics of non-rational consumer behavior. The utility tree is constructed by the approximate nonparametric method.

Technological changes on the macroeconomic level—mathematical modeling

We consider two of the technological changes on the macroeconomic level. The first type is due to changes of addresses of mutual deliveries between producers and the second type is due to technological progress.