Optimal Growth Path Research Articles

Team Production and the Dynamics of the Firm

This paper discusses the behavior of the firm when its nature as a nonmarket allocational device (a team) is taken into account. The activities of the firm are divided into team formation and team production in a dynamic model of the competitive firm. The determinants of stationary equilibrium and the optimal growth path are analyzed. Impact and long-run effects of parameter changes are also derived. It is concluded that textbook results largely continue to hold in stationary equilibrium, but that significant modifications may have to be made with respect to conjectures regarding the short-run behavior of growing firms.

On optimal steady states of n-sector growth models when utility is discounted

This paper contains results on local and global stability of n-sector growth models when utility is discounted mostly for small rates of discount. It is well known that when future utility is not discounted one can prove precise results about optimal steady states (OSS's) under fairly general assumptions. In particular, existence, uniqueness, and turnpike properties have been established by several authors. The counter examples presented by Kurz, Sutherland, and Weitzman, however, show that when utility is discounted, additional assumptions are required to obtain turnpike results. In general, it would be interesting to know how the submanifolds of stability change as δ changes. One hopes that certain conditions on the utility function would be sufficient to “classify” the submanifolds of stability and instability. Such a question is apparently very difficult to answer, but we think that the results obtained here will help in this task. The proof that the turnpike theorem holds for discount factors near one is divided in two parts. First, we prove that optimal paths “visit” neighborhoods of the modified OSS's. Then, we prove that local stability holds for such neighborhoods. In order to show this fact, we must prove that the local “stable manifold” varies continuously with the discount factor. This roundabout method is necessary since our problem is similar to proving uniform continuity with respect to a parameter of solutions of a differential equation in a noncompact interval of time. Other problems analyzed here include uniqueness and continuity of OSS's. We also discuss the relation between the saddle-point property and the local stability of infinite horizon optimal growth paths.

Parametric studies in transportation‐type problems

AbstractIn this paper we present some results in parametric studies on several transportation‐type problems. Specifically, a characterization is obtained for the optimal values of the variables in the problem of determining an optimal growth path in a logistics system. We also derive an upper bound beyond which the optimal growth path remains the same. The results are then extended to the goal programming model and the prespecified market growth rate problem.

Growth with Exhaustible Natural Resources: Efficient and Optimal Growth Paths

The proposition that limited natural resources provide a limit to growth and to the sustainable size of population is an old one. The natural resource that was the centre of the discussion in Malthus' day was land; more recently, some concern has been expressed over the limitations imposed by the supplies of oil, or more generally, energy sources, of phosphorus, and of other materials required for production. Those who predicted imminent doom in the nineteenth century were obviously wrong. Were they simply wrong about the immediacy of catastrophe, or did they leave out something fundamental from their calculations? There are at least three economic forces offsetting the limitations imposed by natural resources: technical change, the substitution of man-made factors of production (capital) for natural resources, and returns to scale. This study is an attempt to determine more precisely under what conditions a sustainable level of per capita consumption is feasible, to characterize steady state paths in economies with natural resources, and to describe the optimal growth path of the economy, in particular to derive the optimal rate of extraction and the optimal savings rate in the presence of exhaustible natural resouces. One of the interesting problems posed by the presence of exhaustible natural resources is that some of the basic concepts of growth theory, such as steady state and natural rate of growth , need to be re-examined. If, for instance, there are two unproduced factors, labour and natural resources, one of which is growing exponentially, the other of which is not growing at all, what is the natural rate of growth ? In conventional economic discussions, the long-run growth rate of the economy is determined simply by the natural rate of growth and is independent of the savings rate. We shall show that in economies with natural resources, efficient growth paths which differ with respect to savings rate also differ, even asymptotically, with respect to the rate of growth. The analysis of optimal growth paths presents certain technical difficulties, because there are two state variables (the stock of capital per man and the stock of natural resources per man) and two control variables (the rate of extraction of natural resources and the savings rate). Fortunately, by the appropriate choice of variables, the qualitative properties of the path can be completely described. Optimal growth paths for economies with only capital or with just natural resources have been examined elsewhere. Typically, a country begins with little capital and hence, in the former models, optimal growth is characterized by increasing consumption per capita. On the other hand, natural resources act much like a capital good; since the stock

Structure, Law of Motion, and Optimal Path of Growth of Complex Urban Systems

This paper presents a mechanism underlying the formation and development of complex urban systems. The problem is approached in the manner of a cybernetic analysis and optimal control theory. The purpose of such an approach is to define in a formal way the structure, the law of motion, and the optimal growth path of complex urban systems. The formal definitions, in turn, may serve as schemes for empirical observations, analysis and generalizations, as well as more detailed models. They may also help to determine overlooked or unsatisfactorily elaborated problems of urban systems and in this way contribute to the formulation of adequate research projects.

Determining optimal growth paths in logistics operations

AbstractThis paper considers a logistics system modelled as a transportation problem with a linear cost structure and lower bounds on supply from each origin and to each destination. We provide an algorithm for obtaining the growth path of such a system, i. e., determining the optimum shipment patterns and supply levels from origins and to destinations, when the total volume handled in the system is increased. Extensions of the procedure for the case when the costs of supplying are convex and piecewise linear and for solving transportation problems that are not in “standard form” are discussed. A procedure is provided for determining optimal plant capacities when the market requirements have prespecified growth rates. A goal programming growth model where the minimum requirements are treated as goals rather than as absolute requirements is also formulated.

On the Relative Stability and Optimality of Consumption in Aggregative Growth Models: A Comment

In an article in this journal, Professor Sengupta investigated the behaviour of optimal growth paths in a model of the Ramsey type [2].1 In our view this article is marred by flaws, the most serious of which is that Sengupta goes through mathematical manipulations without properly defining the problem he is tackling. In particular, the meanings to be attached to the important symbols T, t--T, and lim, are t-.T not clear. Furthermore, whatever meanings one gives to them, inconsistencies remain in Sengupta's analysis. There are also some errors in the mathematics. In the following we examine the first three sections (pp. 33-38) of his article. We begin by recapitulating Sengupta's argument. Sengupta's specification of the model is as follows.2 The system maximizes the utility functional3'4

Zur optimalitÄt der Uzawa-Pfade

On the Optimal Growth of Uzawa Paths. — In a number of recent works optimal growth paths are stated by the classical calculus of variations. In this paper we want to discuss the problems which occur in finding optimal growth paths by the help of the classical calculus of variations. You will find two critical aspects for the well-known growth model of Uzawa. First, by the calculus of variations not all possible economic restrictions are fully covered, so that the possibilities of economic interpretation of the results are not guaranteed uniquely. The results may be, for instance, negative. Secondly, the result found by Uzawa is — respecting the given aims — in no case optimal. The same may be stated to the model of Konig. Therefore the calculus of variations is characterized as a problematical method of finding optimal economic growth paths. On the contrary the maximum principle promises by its mere formulation a higher efficiency.

Truncated Decision Rules and Optimal Economic Growth with a Fixed Horizon

ONE OF TIlE MOST interesting characteristics of recent aggregative ino(dels of economic growth is perhaps the specification of optimal growth paths under varying assumptions and conditions defining optiinality. Assuming that an optimal growth path for a fixed horizon can be specified for a developed or an underdeveloped economy, difficult decision problems of macro-dyniamic policy-making arise, when appropriate governmental actions are needed to secure the convergence of the observed growth path to the optimal one. The solutions of these decision problems by simple mathematical models are considered here with special emphasis on their uncertainty and dynamic aspects. The implications of alternative preference functions are analyzed here to show how optimal decision rules can be derived under various orders of truncation2, where truncation refers to the (degree of incomplete sp)ecification of the probability distribution of the random elements of the problem, i.e., the risk preference function (or disutility function) of the problem. Our approach would be to analyze the concept of 'a linear decision rule' developed by Theil [71 and emphasized recently by Holt [6] who used a quadratic risk function in a simple aggregative Keynesian model based on the multiplicr, the income identity and private investment as the only random variable. For illustrative purposes, we would use the principle of optimality of Pontryagin [9] as a generalization of the extremum conditions of variational calculus to derive the optimal time

On the Relative Stability and Optimality of Consumption in Aggregative Growth Models

Since Ramsey's pioneering work2 on the optimality of savings over time, a number of attempts have been made in the recent economic literature to specify the implications of optimal growth paths of consumption and income in the framework of aggregative economic models of growth. Very recently Professor Goodwin,3 by using a Bernoullitype logarithmic utility function, derived some interesting policy implications of the Ramsey-type optimal growth path, where optimal growth is defined as the maximum of welfare over a finite period, welfare being taken as per capita real consumption valued in some manner. By using more or less the same sort of logarithmic valuation function but in a more general context of an inter-sectoral model, we investigate here the conditions of relative stability of the optimal path of consumption, when the time horizon of planning, although finite, is very large. It is shown that our conditions of relative stability are more general than those derived by Uzawa4 and others5 who have developed a bi-sector model of growth on the same pattern as the Harrod-Domar model. We also examine how far the optimality principle of the Ramsey-Goodwin-type models could be extended to a more general situation, where we are required to derive the optimal time path of a control variable, e.g., public investment, when the time paths of private investment and aggregate consumption are derived partly from the observed behaviour relations and partly from other optimizing considerations.

The Optimal Growth Path for an Underdeveloped Economy

Journal Article The Optimal Growth Path for an Underdeveloped Economy Get access R. M. Goodwin R. M. Goodwin Peterhouse, Cambridge. Search for other works by this author on: Oxford Academic Google Scholar The Economic Journal, Volume 71, Issue 284, 1 December 1961, Pages 756–774, https://doi.org/10.2307/2228248 Published: 01 December 1961