Theory Of Optimal Economic Growth Research Articles

НЕЛІНІЙНА МОДЕЛЬ ПОВЕДІНКИ ДВОХ КОНКУРЕНТНИХ ФІРМ

The practical task of economics lies in applying the methods of substantiating its decisions. For economics, the main method is the modeling of economic phenomena and processes and, above all, mathematical modeling, which has been stipulated by the presence of stable MATHEMATICAL METHODS, MODELS AND INFORMATION TECHNOLOGIES IN ECONOMY Issue I (81), 2021 117 quantitative patterns and the possibility of a formalized description of many economic processes. The economic-mathematical model contains a system of equations of linear and nonlinear units that promote a mathematical description of economic processes and phenomena, consists of a set of variables and parameters and serves to study these processes and control them. Dynamic models of the economy describe it in development, as well as provide a detailed description of technological methods of production. Mathematical description of dynamic models is carried out with the use of a system of differential equations (in models with continuous time), difference equations (in models with discrete time), as well as systems of algebraic equations. It is important that the investigation of various economic issues has led to the development of the mathematical apparatus. In linear algebra, productive matrices are caused by the studies of intersectoral balance, whereas mathematical programming arose in the course of researching the optimal plan for the distribution of limited resources. In a similar way, there emerged the theory of economic indices and econometrics, the theory of production functions and the theory of consumption, the theory of general economic balance and social welfare, the theory of optimal economic growth. The paper under studies deals with the dynamic economic behavior of two competing objects, whose mathematical model is a nonlinear nonlocal problem for a system of ordinary differential equations with variable coefficients and argument deviation. The dynamic mathematical model is based on the assumption that the volume of output of both firms is determined by such factors on which output depends linearly. The model under discussion includes nonlinear factors, which describe the level of distrust of the competitors and depend on the time of observations and production volumes in previous moments, because the latter significantly affect the production activities of the firm. Such mathematical models are called time-delayed models.

On variable discounting in dynamic programming: applications to resource extraction and other economic models

This paper generalizes the classical discounted utility model introduced in Samuelson (Rev. Econ. Stud. 4:155–161, 1937) by replacing a constant discount rate with a function. The existence of recursive utilities and their constructions are based on Matkowski’s extension of the Banach Contraction Principle. The derived utilities go beyond the class of recursive utilities studied in the literature and enable a discussion on subtle issues concerning time preferences in the theory of finance, economics or psychology. Moreover, our main results are applied to the theory of optimal economic growth related with resource extraction models with unbounded utility function of consumption.

AN INTERVIEW WITH HIROFUMI UZAWA

AN INTERVIEW WITH HIROFUMI UZAWA

A variational problem relating to the theory of optimal economic growth

A variational problem relating to the theory of optimal economic growth

On Optimal Depletion of Exhaustible Resources: Existence and Characterization Results

IN RECENT YEARS, there has developed a significant literature on the economics of natural resources. A part of this literature is concerned with finding the characteristics of optimal programs for economies with exhaustible resource constraints; that is, of jointly determining the optimal depletion of such resources, and optimal investment in augmentable capital goods. It should be noted that the theory of optimal economic growth, in the form given it by Ramsey [17], and extensively developed by many others, had been primarily concerned with the latter problem. Natural resources were often assumed to be supplied exogenously in given amounts in each period, an approach clearly unsuitable for capturing the essence of problems associated with the optimal use of exhaustible resources. The more recent literature derives its inspiration from Hotelling's classic paper [8], stressing the increasing significance of such resources in production, as they are irrevocably run down. I shall consider, in this paper, a model of intertemporal allocation in which there is a produced good (which can be used for consumption or for further production), and an exhaustible resource (which is essential for production), the total initial stock of which is given. The use of the resource over the (infinite) planning horizon must not exceed this available stock. A planner is assumed to evaluate consumption in each period, in terms of a utility function, and to "maximize" the undiscounted sum of these one-period utilities, to obtain, simultaneously, the optimal depletion of the exhaustible resource, and the optimal investment 2 pattern. This model resembles, in some aspect or other, the frameworks examined by Dasgupta [3], Dasgupta and Heal [4, 5], Solow [18], and Stiglitz [19], to mention only a few. I will address three sets of issues in this framework. First, an interesting problem in the theory of optimal economic growth is to find suitable conditions under which a competitive program is optimal. I will show that a feasible program is optimal if and only if (a) it is competitive,3 and (b) it satisfies the transversality condition, that the value of the capital and resource stocks, at the competitive prices, converges to zero (Theorem 3.1). It follows from this result, that a competitive program is optimal if and only if it satisfies the above-mentioned transversality condition. It should be noted that in traditional models of optimal growth (in which exhaustible resources do not appear as essential factors of production), a feasible program, under certain technological curvature conditions, is shown to be optimal if and only if it is competitive, and the value of input

Economic development and efficiency criteria in the satisfaction of basic needs

This paper aims to give a possible mathematical formalization of economic efficiency in an economy concerned with attaining acceptable levels of per capita consumption of basic goods, subject to economic and social welfare constraints. This model was constructed as an underlying mathematical economic structure of Latin American world model of Fundacion Bariloche. It has been used in a modified form, as a feasibility study of possible growth paths id such underdeveloped economies. The model has a methodological intersection with the theory of optimal economic growth, and that the efficient growth paths are solutions of a constrained optimization problem: they optimize a social welfare criterion satisfying constraints given by a set of differential equations with initial values. In the usual optimal growth models these differential equations represent production-investment-consumption relations through time, formalizing a trade-off between present and future consumption (more present consumption implies less present investment). Here in addition, the admissible paths also satisfy a differential equation which relates the rate of population growth to the level of consumption of basic goods (a proxy standard of living) which introduces further relationships between present and future consumption. The admissible paths are assumed to satisfy additional requirements per capita consumption of all basic goods must be non-decreasing throughout time. This is a welfare consideration for countries with low levels of consumption. A brief description of the dynamics of the model in sequential form can be given as follows. Initial values of production, consumption, and investment in three basic sectors are given (nutrition, housing, health and education), and in a fourth sector of services and consumption foods complementary to the production and consumption of the three basic goods.

Turnpike Properties for the Optimal Use of a Natural Resource

This paper is an attempt to deal rigorously with some fundamental mathematical problems arising from the consideration of natural resources in the neoclassical framework of optimal economic growth. The theory of optimal economic growth initiated by Ramsey [20] and developed by many others has benefited considerably from an optimal control formulation (see for example [1] and [22]). Incidentally it posed two interesting problems which seem to be typical of these economic models. The first one concerns the precise formulation of the Turnpike Property (see Cass [5]), the second one is the definition and the characterization of the optimality when the time interval considered is infinite and the formulation of existence conditions for such an optimal solution (see Koopmans [14]). It appeared that the two problems are linked together through the important role played in both problems by an optimal steady state suitably defined (see for example Gale [9] and McKenzie [17] for discrete time growth models and Haurie [11] for continuous time control systems). In the case of optimal exploitation of a natural resource, the Turnpike Property is well illustrated by the very simple model of optimal fish-harvesting studied by Cliff and Vincent [6] (this model is very close to the one proposed by Plourde [19]). Those authors showed that, for their particular model, any optimal trajectory on a sufficiently large interval would contain a singular arc where the fish population is maintained at its level of maximal yield, this singular arc being independent of the initial and terminal conditions. This result is similar to the turnpike property first presented in DOSSO [8] for efficient programmes of capital accumulation and then extended to the neoclassical optimal growth framework. However no formal proof has been given yet for an extension of the turnpike property to the case of an optimal growth model where both capital and natural resources enter into the production process. A first difference between capital and resource assets is that the former enter as stock inputs in the production process while the latter are often flow inputs. Another difference is that capital depreciates over time and is reproducible through investment while a natural resource is depleted by its use in the economic process and, when replenishable, reproduces itself according to a natural law. In Section 3 this extended turnpike property is proven for a continuous time model and expressed as a bound for the measure of the time spent away from the Von Neumann Set of the economy, a set which always contains the optimal steady state. This is essentially an adaptation of the classical result obtained by McKenzie in a discrete time setting without natural resources. When the Von Neumann set reduces to a

Economic Development and Efficiency Criteria in the Satisfaction of Basic Needs

This paper aims to give a possible mathematical formalization of economic efficiency in an economy concerned with attaining acceptable levels of per capita consumption of basic goods, subject to economic and social welfare constraints. This model was constructed as an underlying mathematical economic structure of Latin American world model of Fundacion Bariloche. It has been used in a modified form, as a feasibility study of possible growth paths id such underdeveloped economies. The model has a methodological intersection with the theory of optimal economic growth, and that the efficient growth paths are solutions of a constrained optimization problem: they optimize a social welfare criterion satisfying constraints given by a set of differential equations with initial values. In the usual optimal growth models these differential equations represent production-investment-consumption relations through time, formalizing a trade-off between present and future consumption (more present consumption implies less present investment). Here in addition, the admissible paths also satisfy a differential equation which relates the rate of population growth to the level of consumption of basic goods (a proxy standard of living) which introduces further relationships between present and future consumption. The admissible paths are assumed to satisfy additional requirements per capita consumption of all basic goods must be non-decreasing throughout time. This is a welfare consideration for countries with low levels of consumption. A brief description of the dynamics of the model in sequential form can be given as follows. Initial values of production, consumption, and investment in three basic sectors are given (nutrition, housing, health and education), and in a fourth sector of services and consumption foods complementary to the production and consumption of the three basic goods.

Optimal control on an infinite time horizon: The turnpike approach

The optimal control problem with infinite horizon is defined. Under assumptions motivated by the theory of optimal economic growth the controllability of the system, the turnpike property, the asymptotic behavior and the existence of optimal trajectories are fully explored.

On sensitivity considerations in the theory of optimal economic growth †

Abstract Kor stabilizing an economic policy, sensitivity considerations are found essential. Various types of sensitivities that we come across in the theory of optimal economie growth are stated and a unified approach of sensitivity reduction in tho Ramsey planning problems is given. An example is provided for illustration.

Intergenerational Equity and Exhaustible Resources

The theory of optimal economic growth, in the form given it by Frank Ramsey and developed by many others, is thoroughly utilitarian in conception. It is utilitarian in the broad sense that social states are valued as a function of the utilities of individuals (individual moments of time, in this case, since individual persons are usually taken as identical and identically treated) with the possibility that a loss of utility to one individual (or generation) can be more than offset by an increment to another. It is also utilitarian in the narrow sense that social welfare is (usually 3) defined as the sum of the utilities of different individuals or generations. Recently the whole utilitarian approach to social choice has come under fundamental attack by John Rawls [9]. One particular view advanced by Rawls concerns me here. He argues, in effect, that inequality in the distribution of wealth or utility is justified only if it is a necessary condition for improvement in the position of the poorest individual or individuals. In other words, if social welfare, W, is to be written as a function of individual utilities U1, ..., U,, then Rawls argues for the particular function W = min (U1, ..., Un), so that maximizing social welfare amounts to maximizing the smallest Ui.4 This welfare function is sensitive only to gains and losses of utility by the poorest person. A Theory of Justice contains a section 5 explicitly devoted to equity between generations, i.e. the question that arises in the theory of optimal capital accumulation. Remarkably, the one thing this chapter does not do is to advocate unequivocally the max-min criterion espoused elsewhere in the book. In this context Rawls settles for such ambivalent statements as the following: . . . the question of justice between generations.. . subjects any ethical theory to severe if not impossible tests.... I believe that it is not possible, at present anyway, to define precise limits on what the rate of savings should be. How the burden of capital accumulation and of raising the standard of civilization is to be shared between generations seems to admit of no definite answer. It does not follow, however, that certain bounds which impose significant ethical constraints cannot be formulated.... Thus it seems evident, for example, that the classical principle of utility leads in the wrong direction for questions of justice between generations. . . . Thus the utilitarian doctrine may direct us to demand heavy sacrifices of the poorer generations for the sake of greater advantages for later ones that are far better off. But this calculus of advantages which balances the

Optimal Capital Accumulation in a Polluted Environment

In The Costs of Economic Growth [8], Dr. Mishan criticizes the importance attached to economic growth by politicians, businessmen, and economists. He contends that these groups generally overlook the spillover effect of the growth process-such as environmental pollution. This is nowhere more apparent than in the modem theories of economic growth (see for example [2]). These theories implicitly assume no wastes are produced by the economic process, or, alternatively (and more likely), that if any wastes are generated they can be disposed of at no cost to the community. It is naive to think that no wastes are produced and fairly obvious that the free disposal assumption of the neoclassical growth model is not satisfied in the real world. Accordingly, any theory of optimal economic growth that does not take into account these spillover effects cannot claim to be complete. This paper will investigate the effects of introducing pollution explicitly into the neoclassical growth model. The model is formulated in Section I. Section II contains the solution, while Section III summarizes the discussion.