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Mathematical Methods of Optimization and Economic Theory
Abstract
Problems of optimization are pervasive in the modern world, appearing in science, social science, engineering, and business. Recent developments in optimization theory, especially those in mathematical programming and control theory, have therefore had many important areas of application and promise to have even wider usage in the future.
This book is intended as a self-contained introduction to and survey of static and dynamic optimization techniques and their application to economic theory. It is distinctive in covering both programming and control theory. While book-length studies exist for each topic covered here, it was felt that a book covering all these topics would be useful in showing their important interrelationships and the logic of their development. Because each chapter could have been a book in its own right, it was necessary to be selective. The emphasis is on presenting as clearly as possible the problem to be treated, and the best method of attack to enable the reader to use the techniques in solving problems. Space considerations precluded inclusion of some rigorous proofs, detailed refinements and extensions, and special cases; however, they are indirectly covered in the footnotes, problems, appendices, and bibliographies. While some problems are exercises in manipulating techniques, most are teaching or research problems, suggesting new ideas and offering a challenge to the reader. Most chapters contain a bibliography, and the most important references are indicated in the first footnote of each chapter. The most important equations are numbered in bold face type.
... If used in an optimization model, PMP can replicate the observed resource use behavior actually seen. This behavior comes from the "first order necessary conditions" (Intriligator, 2002) for optimal resource use. ...
Successful climate adaptation needs to sustain food, water, and energy security in the face of elevated carbon emissions. Hydroeconomic analysis (HEA) offers considerable potential to inform climate adaptation plans where water is an important element of economic activity. This paper's contribution is to identify how HEA can inform climate adaptation plans by minimizing economic costs of responding to climate induced changes in water supplies. It describes what HEA is, why it is important, how researchers implement it, who has made significant contributions, and places where it has informed policy debates. It also describes future directions for the use of HEA to guide climate adaptation.
... The two most important factors of production are capital and labor, also the technology change alters the production function (Solow, 1956). We consider the functions that satisfy the following conditions (Intriligator, 2002): ...
In recent years, eco-efficiency assessment has proven to be an effective tool to reduce the environmental damages of agricultural activities while preserving their economic sustainability. Hence, this paper aims to assess the eco-efficiency of a sample of 148 beef cattle farms operating in the extensive livestock system of Central Italy. The analysis is based on Farm Accountancy Data Network (FADN) economic data in the year 2020 and includes, as environmental pressures, farm expenditure for the use of fuels, electricity and heating, and fertilizers. A two-stage approach was implemented: in the first stage, an input-oriented DEA model including slack variables was used to quantify farm eco-efficiency scores and determine the polluting inputs’ abatement potentials. In the second stage, the influence of possible influencing factors on eco-efficiency scores was tested using a regression model for truncated data. The analyzed farms were found to be highly eco-inefficient, as they could abate their environmental pressures, on average, in a range from 56% to 60% while keeping the value of their global production constant. Fertilizers and fuel consumption were identified as the least efficiently operating inputs, with potential reductions in terms of the related expenditures fluctuating between 9% and 42%. Farms showing a high-intensity livestock system, a low labor intensity, and a larger farm area were recognized as the most eco-efficient. Environmental and animal welfare subsidies were found to not affect eco-efficiency, while a negative influence was estimated for a single farm payment, which does not seem to be an incentive mechanism for farms to operate efficiently.
The paper aims to develop a mathematical model of consumer demand observed over a certain number of years. Compression (reduction) of a large amount of data in a mathematical model to compact mathematical objects—matrices and their eigenvectors—allows one to make certain conclusions about the general set of consumers in several markets at once. The basic mathematical apparatus used is Allen’s approach, also known as the best average percentage. The research is based on one of the varieties of the ordinary least squares method (OLS). The paper considers the possibility of detecting the presence of periodicity in consumer demand. A mathematical criterion for accepting or rejecting assumptions about periodicity is also proposed. The proposed meta-mathematical model allows us to process large data sets on consumer demand of past periods and predict the periodicity of demand.KeywordsDemandHomogeneityFrequencyRandom errors of observationMatrix traceEigenvaluesJEL ClassficationsC020C3
In this paper, we perform a sensitivity analysis for the maximal value function, which is the optimal value function for a parametric maximization problem. Our aim is to study various subdifferentials for the maximal value function. We obtain upper estimates of Fréchet, limiting, and horizon subdifferentials of the maximal value function by using some sensitivity analysis techniques sophisticatedly. The derived upper estimates depend only on the union of all solutions and not on its convex hull or only one solution from the solution set. Finally, we apply the derived results to develop some new necessary optimality conditions for nonconvex minimax problems. In the nonconvex-concave setting, our Wolfe duality approach compares favorably with the first-order approach in that the necessary condition is sharper and the constraint qualification is weaker.
Funding: L. Guo was supported by the National Natural Science Foundation of China [Grants 72131007, 72140006, and 12271161] and the Natural Science Foundation of Shanghai [Grant 22ZR1415900]. J. J. Ye was partially supported by the Natural Sciences and Engineering Research Council of Canada. J. Zhang was supported by the National Natural Science Foundation of China [Grant 12222106], the Shenzhen Science and Technology Program [Grant RCYX20200714114700072], and the Guangdong Basic and Applied Basic Research Foundation [Grant 2022B1515020082].
Optimization is about finding the best available object with respect to an objective function. Mathematics and quantitative sciences have been highly successful in formulating problems as optimization problems, and constructing clever processes that find optimal objects from sets of objects. As computers have become readily available to most people, optimization and optimized processes play a very broad role in societies. It is not obvious, however, that the optimization processes that work for mathematics and abstract objects should be readily applied to complex and open social systems. In this paper we set forth a framework to understand when optimization is limited, particularly for complex and open social systems.
In this paper a Feynman-type path integral control approach is used for a recursive formulation of a health objective function subject to a fatigue dynamics, a forward-looking stochastic multi-risk susceptible-infective-recovered (SIR) model with risk-group's Bayesian opinion dynamics toward vaccination against COVID-19. My main interest lies in solving a minimization of a policy-maker's social cost which depends on some deterministic weight. I obtain an optimal lock-down intensity from a Wick-rotated Schrödinger-type equation which is analogous to a Hamiltonian-Jacobi-Bellman (HJB) equation. My formulation is based on path integral control and dynamic programming tools facilitates the analysis and permits the application of algorithm to obtain numerical solution for pandemic control model.
Following the dividend flexibility hypothesis used by DeAngelo and DeAngelo (J Financ Econ 81, 227–254, 2006), Blau and Fuller (J Corp Financ 14:133–152, 2008), and others, we theoretically extend the proposition of DeAngelo and DeAngelo’s (J Financ Econ 81, 227–254, 2006) optimal payout policy in terms of the flexibility dividend hypothesis. We also introduce growth rate, systematic risk, and total risk variables into the theoretical model. In addition, based upon Lee and Lee (2021), we discuss the implication of the existence of optimal payout ratio in financial analysis and decision for a company. To test the theoretical results derived in this chapter, we use data collected in the USA from 1969 to 2009 to investigate the impact of growth rate, systematic risk, and total risk on the optimal payout ratio in terms of the fixed-effect model. We find that based on flexibility considerations, a company will reduce its payout when the growth rate increases. In addition, we find that a nonlinear relationship exists between the payout ratio and the risk. In other words, the relationship between the payout ratio and risk is negative (or positive) when the growth rate is higher (or lower) than the rate of return on total assets. Our theoretical model and empirical results can therefore be used to identify whether flexibility or the free cash flow hypothesis should be used to determine the dividend policy.
Supply chain development has globally increased the importance of rail transport systems. This importance is mainly attributed to high speed, safety, reliability, lower cost, and being eco-friendly compared to road transportation. This chapter examines the overview of the rail freight network, its role in the supply chain, scientific literature, and current concerns. Rail network concerns are investigated considering six essential elements: environment, cost, optimization, operation, planning, and safety and resilience. Furthermore, a comprehensive causality network is developed to manage the railway network effectively. Finally, future directions, opportunities, and challenges in this domain are presented.
Paul A. Samuelson’s (1966) capitulation with his Austrian model during the so-called Cambridge controversy on the phenomenon of re-switching of techniques in capital theory had implications not only in pointing at the supposed internal contradiction of the marginal theory of production and distribution but also in the pursuit of vested interests in the academic and political world to this day. The present paper is aimed at demonstrating that Samuelson’s capitulation was logically groundless from the point of view of the economic theory of production by considering the interest rate in the role of the real price of capital goods instead of the stationary real rental prices as correctly suggested by Leon Walras, Knut Wicksell, John Hicks, and others. Because of the non-linear effects of the interest rate on relative input prices, the Sraffian re-switching of techniques noted in the range of the interest rate always disappears in the space of the corresponding real input prices in the stationary equilibrium.
The article deals with the problem of integrating logical-heuristic and economic-mathematical models of coalition games concerning the distribution of limited investment resources of the state-investor between the performers (contractors) of competing railway projects. Using the example of the latter, which are planned to be implemented in the east of the country in the future, the task is to find such a distribution of limited investment resources that would ensure the maximum possible stability of a particular system of construction contractors. For these purposes, it is proposed to use a modified Shapley algorithm and a software product developed in the SGUPSE that works on expert information. The experimental calculation shows the efficiency of the created system and its usefulness in finding the most allocatively efficient allocation of resources between coalitions of action (contractors).KeywordsShapley vectorCoalition gamesLarge-scale investment projectsRailway linesLogical-heuristic modelAllocative efficiency
In this paper, we consider the application of a modern approach to solving nonlinear optimal control problems using as an example a relevant problem applied to the Russian insurance market. The actuarial problem is examined as an optimal control problem with dynamic constraints on the control. The approach for solving optimal control problems is based on R. Gabasov’s adaptive method. The linear problem is reduced to an interval linear programming problem and the linear programming problem is solved by the adaptive method.
In recent years, aquaculture has become a global practice and is now widely adopted by the majority of countries in response to the growing demand for seafood products. Yet, as the wild fisheries continue, careful planning is required to ensure that farmed fish are introduced to the market in an environmentally sound manner. In this paper, we will develop sufficient conditions for overfishing to occur when fishermen are faced with a future aquaculture installation allowing the commercialization of substitutes for wild marine products. By considering different scenarios in a two-stage dynamic model, we show that depending on the model parameters, aquaculture activities can harm the marine biodiversity and becomes an incentive for the over-exploitation of the resource. Moreover, when the aquaculture is owned by the only fisherman in the industry, this situation may prevail.KeywordsDynamic gamesFisheriesTwo-stage gamesAquacultureBlue paradox
A duopoly model with a linear demand function and nonlinear cost functions of agents is considered. The game with the multilevel Stackelberg leadership is investigated. We analyze conjectural variations, i.e., the agent’s assumption about changes in the counterparty’s actions, which optimize the latter’s utility function. For an arbitrary Stackelberg leadership level, the formula for calculating the conjectural variations of agents is derived. The main insights are as follows: (1) the variations depend not only on the leadership level, but also on the product of the cost functions concavity/convexity indicators; (2) if at least one of agents has the concave cost function, then the variations can be not only negative, but also positive, and are not limited in absolute value, i.e., the bifurcations can occur.KeywordsDuopolyStackelberg gamePower cost functionMulti-level leadership
The paper is devoted to the analysis of behavior of equilibrium trajectories for game dynamic systems arising in solution of bimatrix games. At the first stage, the approach is considered based on the ideas of guaranteed strategies in the sense of N.N. Krasovskii. In the framework of guaranteed solutions, we propose algorithms for constructing the value functions, positional strategies and equilibrium trajectories using the definition of the dynamic Nash equilibrium. At the second stage, we analyze equilibrium trajectories of the replicator dynamics relating to the theory of evolutionary games. At the third stage, we examine the dynamic system generated by the strategies of best replies similar to the Cournot model. The comparison is carried out for the objective indices of the equilibrium trajectories of all three dynamic systems. It is shown that the characteristics of the trajectories of the dynamic Nash equilibrium are better than the properties of the trajectories of the replicator dynamics or the best reply dynamics. In addition, the numerical experiments are implemented for the so-called mixed dynamics in which the first player uses the guaranteed strategy and the strategy of the second player is formed either by the replicator formulas or by the best reply dynamics. The simulation results for the mixed dynamics demonstrate that the values of players’ payoff functionals in the final points of trajectories are better in comparsion with the indices for trajectories of the replicator dynamics and the best reply dynamics and even better than at the final point of the dynamic Nash equilibrium.
The exponential growth in the number of flash flood events is a global threat, and detecting a flood-prone area has also become a top priority. The flash flood-susceptibility mapping can help to mitigate the worst effects of this type of risk phenomenon. However, there is an urgent need to construct precise models for predicting flash flood-susceptibility mapping, which can be useful in developing more effective flood management strategies. In this present research, support vector regression (SVR) was coupled with two meta-heuristic algorithms such as particle swarm optimization (PSO) and grasshopper optimization algorithm (GOA), to construct new GIS-based ensemble models (SVR–PSO and SVR–GOA) for flash flood-susceptibility mapping (FFSM) in the Gandheswari River basin, West Bengal, India. In this regard, 16 topographical and environmental flood causative factors have been identified to run the models using the multicollinearity (MC) test. The entire dataset was divided into 70:30 for training and validating purposes. Statistical measures including specificity, sensitivity, PPV, NPV, AUC–ROC, kappa and Taylor diagram have been employed to validate adopted models. The SVR-based factor importance analysis was employed to choose and prioritize significant factors for the spatial analysis. Among the three modeling approaches used here, the ensemble method of SVR–GOA is the most optimal (specificity 0.97 and 0.87, sensitivity 0.99 and 0.91, PPV 0.97 and 0.86, NPV 0.99 and 0.91, AUC 0.951 and 0.938 in training and validation, respectively), followed by the SVR–PSO (specificity 0.84 and 0.84, sensitivity 0.87 and 0.86, PPV 0.85 and 0.82, NPV 0.87 and 0.87, AUC 0.951 and 0.938 in training and validation, respectively) and SVR (specificity 0.80 and 0.77, sensitivity 0.93 and 0.89, PPV 0.82 and 0.77, NPV 0.91 and 0.89, AUC 0.951 and 0.938 in training and validation, respectively) model. The result shown that 40.10 km² (10.99%) and 25.94 km² (7.11%) areas are under very high and high flood-prone regions, respectively. This produced reliable results that can help policymakers at the local and national levels to implement a concrete strategy with an early warning system to reduce the occurrence of floods in a region.
The production correspondence associated with a technology maps every input vector into the set of output vectors that may be obtained by means of those inputs. The cost function induced by a production correspondence assigns to every pair consisting of a vector of input prices and an output vector the smallest possible cost that has to be paid for achieving that output vector under those input prices. Our main result characterizes cost functions and establishes a bijection between them and the set of quasi-concave production correspondences. In the framework of this bijection, we characterize co-radiant, as well as positively homogeneous, production correspondences. We also study demand correspondences.
In this chapter, the compartmental approach is applied to build a macroeconomic model characterized by countries. We consider a total of N countries that are subdivided into three compartments according to their economic status: D(t) denotes the compartment of developing countries at time t, E(t) stands for the compartment of emerging countries at time t, while A(t) represents advanced countries at time t. The model describes the process of economic development and includes the notion of openness through collaborations between countries. Two delays appear in this model to describe the average time necessary for collaborations between countries to become efficient for their development process. Our model represents the different stages of development. It further gives the conditions under which a country can change its economic status and demonstrates the short-term positive effect of openness on economic growth. In addition, we investigate bifurcation by considering the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations from a positive equilibrium. Numerical simulations are provided in order to illustrate the theoretical part and to support discussion.
A duopoly model with a linear demand function and non-linear cost functions of agents is considered. We study a game with multi-level Stackelberg leadership. This game arises in the case of an oligopoly in the telecommunications market that connects consumers, services supply equipment, and management of the telecom operators, therefore, the market is related to cyber-physical systems. In this case, the decision-making mechanism of the management is based on the analysis of mutual assumptions of the operators about the possible strategies of competitors, taking into account the technical characteristics of the equipment. Conjectural variations (i.e., changes in the counterparty’s actions assumed by the agent that optimize a utility function of the latter) are analyzed. Formulas for calculating the conjectural variations of agents are derived. Signs and boundaries of the variations for an arbitrary Stackelberg leadership level are investigated.
This contribution evokes Orio Giarini’s courage to think ‘outside the box’. It proposes a practical way to bridge the gap between risk (where probabilities of occurrence are fully known) and uncertainty (where these probabilities are unknown). However, in the context of insurance, neither extreme applies: the risk type of a newly enrolled customer is not fully known, loss distributions (especially their tails) are difficult to estimate with sufficient precision, the diversification properties of a block of policies acquired from another company can be assessed only to an approximation, and rates of return on investment depend on decisions of central banks that cannot be predicted too well. This contribution revolves around the launch of an innovative insurance product, where the company has a notion of whether a favourable market reception is more likely than an unfavourable one, of the chance of obtaining approval from the regulatory authority and the risk of a competitor launching a similar innovation. Linear partial information theory is proposed and applied as a particular practical way to systematically exploit the imprecise information that may exist for all of these aspects. The decision-making criterion is maxEmin, an intuitive modification of the maximin rule known from games against nature.
Parallelization schemes on many-core architectures, in this case, CUDA and OpenMP, are used to accelerate and improve the accuracy of adaptive multidimensional integration algorithms. The one-dimensional Gauss–Kronrod adaptive method is generalized to 3, 4, 5 and 6 dimensions. The implementation of the traditional tensor product construction of the grid and weights for multidimensional integration is revisited and reformulated taking advantages of the multi and many-core architectures. Tests performed in a set of benchmark functions show that the algorithm is numerically accurate, with accelerations as high as 800X in CUDA and 300X in the OpenMP implementation both compared to a sequential multidimensional integration algorithm.
In the paper, a problem of socio-economic system management, as exemplified by one of the regions of the Russian Federation, is solved. An algorithm of an optimal control using an index method for building a transitional period and with an ability to take into account growth rates in the engineering and social fields is created. Terms of reference are given on the basis of a regional economic system macro-level model, with production capital and human capital viewed as development factors. A human capital modeling underlying hypothesis is an assumption that it is built on three main aggregative premises: healthcare, education and culture. Including human capital factor in the macro model and an ability to consider progress growth rates are two distinctive features of a given management problem statement. The problem solving algorithm comprises two steps: building the quasi magistral itself and figuring an optimized trajectory which would propel and economic system to the quasi magistral. A gateway period is a transitional period where an optimal control is built using an index method. A transitional period can be diminished by speeding up progress growth rates in the social and engineering fields. In this paper, a two-stage way to build an optimized investment distribution is used upon a model which contains more than to factors for the first time. In this case, a quasi-stationary state is an optimized balanced growth half line. As a statistical base for the analysis, a demographic data and an Udmurt Republic human capital and production capital investment volume data are used. For an unknown parameters’ identification, a period of years 1998–2018 is drawn upon. Optimized investment rates which enable an economic system to reach balanced growth trajectory by the year 2025 are calculated. The proposed method can be used for an optimal control of socio-economic systems, as well as for comparing and estimating their growth rates in the social and engineering fields.
The processes of forced internal migration, which became significant in 2014 as a result of the armed conflict in the east of the country, caused significant demographic and social changes in the regions performance. Particularly large changes have been taken place in the areas directly adjacent to the joint forces operation zone. The study is devoted to the research of the impact of the described processes on certain aspects of social and economic security of the regions. Impact assessment was performed on the basis of cluster analysis. In particular, the author constructed a neural network such as the Kohonen map. The model divided the neural sample from 25 regions (24 regions and the city of Kyiv) into six clusters according to the level of four indicators of social and economic security. This allowed assessing the impact of forced internal migration on some aspects of social and economic security of the regions. Based on the obtained map, it has been depicted that Donetsk and Luhansk regions, which directly border the joint forces operation zone, had a dramatic increase in the demographic burden and unemployment rate during the study period. The obtained results allowed assessing the impact of the processes of forced internal migration on the dynamics of certain indicators of social and economic security of the territories.
The game-theoretic problem of choosing optimal strategies for oligopoly market agents with linear demand functions and nonlinear cost functions is considered. Necessary conditions for the existence of a solution of a system of nonlinear equations with power functions are established. The system of equations for the optimal responses of agents is linearized by expanding the power functions in Taylor series. As a result, the linearized system depends on the vector of linearization parameters, and the calculation of game equilibria is reduced finding fixed points of nonlinear mappings. The deviations of the approximate equilibrium from the exact solution are investigated. Analytical formulas for calculating equilibria in the game of oligopolists under an arbitrary level of Stackelberg leadership are derived. Analysis of duopoly and tripoly demonstrates that the game equilibrium is determined by two factors as follows. First, the concavity of the agent’s cost function (the positive scale effect) leads to an increase in his payoff compared to the agents with convex cost functions (the negative scale effect). Second, the agent’s payoff increases if he is a Stackelberg leader; however, the advantage of his environment by the type of cost function reduces the effect of the second factor.
The purpose of this work is to consider some basic mathematical modeling aspects of economic processes. Economic and mathematical modeling, which is an economic object or process mathematical description, is particularly relevant in management processes. Comprehensive assessment of all important conditions of economic facility operation, management parameters, consequences of decisions made significantly reduces risks and financial losses. The problem is that mathematics does not work with real objects, but with their mathematical models. The problem’s mathematical formulation is only half of the success in solving it. The complexity consists in an exception of excess, unnecessary details. Important conditions should be preserved and the problem formulated as a standard model. This makes it possible to identify process patterns and to ensure effective application of this model in practice. By way of illustration, the author presents some economic problems solved by special mathematical techniques means.
In standard models of spatial harvesting, a resource is distributed over a continuous domain with an agent who may harvest everywhere all the time. For some cases though (e. g., fruits, mushrooms, algae), it is more realistic to assume that the resource is located at a fixed point within that domain so that an agent has to travel in order to be able to harvest. This creates a combined travelling-and-harvesting problem where slower travel implies lower travelling cost and, due to a later arrival, a higher abundance of the resource at the beginning of the harvesting period; this, though, has to be traded off against less time left for harvesting. Possible bounds on the controls render this problem even more intricate. We scrutinise this bioeconomic setting using a two-stage optimal control approach, and find that the agent economises on the travelling costs and thus avoids to arrive at the location of the resource too early. More specifically, the agent adjusts the travelling time so as to be able to harvest with maximum intensity at the beginning and end of the harvesting period, but may also find it optimal to harvest at a sustainable level where the harvesting and the growth rate of the stock coincide in an intermediate time interval.
The article is devoted to the construction and analysis of the simplest mathematical model illustrating the Giffen’s effect and the reasons and conditions for its manifestation. We analyse erroneous, but widely spread, ideas about Giffen’s goods as a good, the demand for which grows due to its relative cheapening against the rising prices for all consumed goods. Under the model it is shown that any good can be both valuable and of little value, at least if it has a more expensive substitute. This property is not an intrinsic and inalienable property of one or another good. The certain property is given to any good by a specific consumer due to its personal preferences and under the influence of existing prices. Inferior good, including such, the consumption of which is available only to an individual with a high level of income, may turn out to be a product of Giffen. Therefore, the consumption of Giffen goods cannot be considered as evidence of a low standard of living for the consumer. Because of the solution of the standard task on the consumer choice, it is shown that the increase in demand for an inferior good when its price is growing, which is the essence of the Giffen paradox, is the result of optim. It is shown that for the manifestation of the Giffen effect it is necessary that the amount of funds allocated by the consumer for acquiring low value good and its more expensive substitute gets into a certain rather narrow range of values.
An oligopoly market with a Stackelberg leader (leaders) and the reflexive behavior of market participants (agents) is considered; for this market, the problem of determining equilibria in the case of incoincident reflexion ranks and different marginal and constant costs of agents is studied. A reflexive game model for a duopoly market is developed and formulas for calculating informational equilibria under incoincident reflexion ranks and different marginal and constant costs of agents are obtained. As is demonstrated below, the advanced (lagged) reflexion of one agent compared to the counteragent affects the intensity of competition in the oligopoly market, making non-uniform the payoff distribution between the agents in favor of the reflexive leader.
A set of models of sequentially growing complexity for enterprise multicriteria decision-making (output planning, borrowing/granting funds, investments in efficiency improvement and production capacity increase) is considered. Simulation results are presented, and also the development and application of system optimization methods to the modeling of production and economic activities of enterprises are discussed.
Economic theory usually focuses on the optimization of a single criterion subject to a set of rigid constraints. Nevertheless, many decisions in economics require to find a balance between multiple and possibly conflicting criteria to obtain a more realistic representation of actual world problems. In addition, there is a lack of a formal specification of multicriteria economic reasoning. In an attempt to strengthen the links between economics and operations research, we propose a formal approach that underpins an economy in which decisions are made by applying multiple criteria methods. From the analysis of the intersection between economic problems and multiple criteria decision making, we find theoretical patterns underlying multicriteria economics. This formal specification allows us to introduce a set of principles that facilitates the development of more realistic multicriteria economics policies.
The paper is devoted to the construction and investigation of mathematical models of economic processes in a local product market. The problem of optimization of prices at outlets of an autonomous network of wholesale under additional restrictions is in focus. The mathematical model of this problem belongs to the class of linear problems of vector optimization. The main properties of the multicriteria problem are studied. An optimal plan is defined. The necessary and sufficient conditions for optimality are established. The theorem of the existence and uniqueness of the optimal plan is formulated. A finite iterative procedure for the problem solution is developed on the base of the obtained theoretical results. The suggested numerical algorithm is based on specific variations of model parameters. The results are illustrated by examples of numerical solutions of some intuitive economic problems with using model data.
The paper discusses various approaches to solving nonlinear optimal control problems. Of all such approaches, we chose the two most characteristic. The first one uses sufficient conditions of optimality in the form of Hamilton–Jacobi–Bellman equations and the corresponding numerical method. The second is based on the reduction of optimal control problem to interval linear programming problem and finding a solution using the Gabasov’s adaptive method. The main goal is to compare the capabilities of these methods within a specific problem of optimal control. As an application, we consider the problem of constructing optimal control in a nonlinear model of macroeconomic growth with nonlinear dynamical constraints. Comparative analysis of these two approaches and corresponding numerical simulation are presented.
A recently developed model of collective intelligence (CI) has been proposed to have the capacity for general collective intelligence (GCI), that is, the capacity for general problem-solving ability that can be reapplied across any domain. This paper explores the relationship between this model of GCI and a model proposed to describe existing CI solutions that are conventional in the sense of accomplishing a single function. The properties required for GCI in this model, and how they make this model unique from other approaches to CI, as well as the implications of these differences, are also explored. In addition, the implications of GCI are explored in terms of the capacity to drive societal impact at transformative scale, where that impact is suggested not to be reliably possible with other approaches to CI.
The industry character in a number of Russian cities historically remains monopolar. According to the authors, the way of their development can be chosen by assuming that the distribution by type of production for the whole region should be close to statistically normal. Diversification based on the proposed mathematical algorithm contributes to solving a topical problem of reducing the number of depressive monocities as concentrators of social tension in Russian society. © 2014, Russian Presidental Academy of National Economy and Public Administration. All rights reserved.
Asymptotic behavior of the value function is studied in an infinite horizon optimal control problem with an unlimited integrand index discounted in the objective functional. Optimal control problems of such type are related to analysis of trends of trajectories in models of economic growth. Stability properties of the value function are expressed in the infinitesimal form. Such representation implies that the value function coincides with the generalized minimax solution of the Hamilton–Jacobi equation. It is shown that that the boundary condition for the value function is substituted by the property of the sublinear asymptotic behavior. An example is given to illustrate construction of the value function as the generalized minimax solution in economic growth models.
A detailed discussion of the mathematical foundations of input–output analysis is given in this chapter. The basic data format used in economic input–output models is discussed first, leading to the basic formulation and its solution using the matrix inversion approach. The use of the basic input–output model for key sector analysis is illustrated with the aid of an example. Applications of the input–output equations are depicted using spreadsheets, which provide an ideal foundation to understand the LINGO codes in subsequent chapters. Then, extensions involving mathematical programming, regional input–output models, and physical input–output models are considered; these variants are also illustrated with examples.
Properties of the value function are examined in an infinite horizon optimal control problem with an integrand index appearing in the quality functional with a discount factor. The properties are analyzed in the case when the payoff functional of the control system includes a quality index represented by an unbounded function. An upper estimate is given for the growth rate of the value function. Necessary and sufficient conditions are obtained to ensure that the value function satisfies the infinitesimal stability properties. The question of coincidence of the value function with the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation is discussed. The uniqueness of the corresponding minimax solution is shown. The growth asymptotic behavior of the value function is described for the logarithmic, power, and exponential quality functionals, which arise in economic and financial modeling. The obtained results can be used to construct grid approximation methods for the value function as the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation. These methods are effective tools in the modeling of economic growth processes.
We study the sensitivity of a functional on the basis of the macroeconomic model. This analysis is a calculation of the derivative with respect to the parameters of the functional characterizing the optimal trajectory. To solve this problem, we apply an approach using conjugate functions and bring the results down to concrete computations. As the model we use a neoclassical model of optimal economic growth and estimate the sensitivity with the growth rate of civilian labor force of national economies in three European countries. Our results can be recommended for analysis and practical use by the relevant authorities. Since the ultimate goal of modeling is to consider feasible alternatives when making decisions, such analysis can be useful.
One of the problems faced by a firm that sells certain commodities is to determine the number of products that it must supply in order to maximize its profit. In this article, the authors give an answer to this problem of economic interest. The proposed problem is a generalization of the results obtained by Stirzaker (Probability and Random Variables: A Beginner’s Guide, 1999) and Kupferman (Lecture Notes in Probability, 2009) where the authors do not present a situation where the sale of a quantity from some commodities is constrained by the marketing of another.
This paper considers models of evolutionary non-zero-sum games on the infinite time interval. Methods of differential game theory are used for the analysis of game interactions between two groups of participants. We assume that participants in these groups are controlled by signals for the behavior change. The payoffs of coalitions are defined as average integral functionals on the infinite horizon. We pose the design problem of a dynamical Nash equilibrium for the evolutionary game under consideration. The ideas and approaches of non-zero-sum differential games are employed for the determination of the Nash equilibrium solutions. The results derived in this paper involve the dynamic constructions and methods of evolutionary games. Much attention is focused on the formation of the dynamical Nash equilibrium with players strategies that maximize the corresponding payoff functions and have the guaranteed properties according to the minimax approach. An application of the minimax approach for constructing optimal control strategies generates dynamical Nash equilibrium trajectories yielding better results in comparison to static solutions and evolutionary models with the replicator dynamics. Finally, we make a comparison of the dynamical Nash equilibrium trajectories for evolutionary games with the average integral payoff functionals and the trajectories for evolutionary games with the global terminal payoff functionals on the infinite horizon.
We consider the problem of finding equilibria in games with three agents on an oligopolic market with a linear demand function and nonlinear agent cost functions. Under strategic reflexion of the agents regarding the presence of a Stackelberg leader (leaders) of the first and second levels, we obtain expressions for information equilibria. Modeling real agent costs and demand functions of the Russian telecommunication market has allowed us to construct a set of information equilibria which we have compared with parameters of the real market and showed the presence of reflexion of the first and second ranks.
Two geopolitical actors implement a geopolitical project that involves transportaion and storage of some commodities. They interact with each other through a transport network. The network consists of several interconnected vertices. Some of the vetrices are trading hubs, storage spaces, production hubs and goods buyers. Actors wish to satify the demand of buyers and recieve the highest possible profit subject to compromise solution principle. A numerical example is given.
In previous chapters, the notion of price was used as an empirical estimate of value of a product. The price is not an intrinsic characteristic of the product as a thing, but it emerges as a result of a bilateral assessment: a producer estimates efforts and expenses necessary to create a thing and a consumer estimates usefulness of that thing for him. The price emerges as a result of an agreement between the producer and consumer, and it thus appears connected with features of behaviour of economic agents. However, this does not mean that price is a subjective quantity; the price of a product exceeds expenses (cost) of manufacture for an amount, which the consumer can pay willingly, so that the attribution of value of a set of products to the production factors is not unreasonable. The relationship between producers and consumers in a process of exchange of products is a market of products. The theory of prices is a theory of the market. In this chapter, the theory of prices is considered for simple schemes that can be described in macroeconomic terms.
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