Optimal Growth Model Research Articles

The world’s productivity distribution and optimal knowledge absorption

An optimal growth model is devised and subject to dynamic analysis. The model is a standard intertemporal utility maximization framework, with a one-dimensional constraint characterizing the pace of productivity growth. In this setting, knowledge acquisition depends not only on the investment in the adoption of existing technologies, but also on the pre-determined shape of the world’s productivity distribution. The configuration of the distribution will determine the probability of successful imitation and, therefore, the extent in which the economy might approach the world’s technology frontier. The dynamic analysis points to the formation of a saddle-path stable equilibrium. The type of distribution is decisive in defining the exact location of the stable trajectory and of the equilibrium point.

A Study on the Growth and Development Characteristics of Lindian Chickens.

As an excellent chicken breed found in a high-altitude zone of northern China, Lindian chickens are characterized by good egg and meat production, strong adaptability, cold tolerance, rough feeding resistance, excellent egg quality, and delicious meat quality. To facilitate the exploitation of the unique qualities of the Lindian chicken, the varying patterns and correlations of various body size and carcass traits of 3-22-week-old Lindian chickens were analyzed in this study. The optimal growth model of these traits was determined by growth curve fitting analysis. The results showed that most traits of Lindian chickens increased steadily with increasing age, and most of them increased rapidly before 10 weeks of age. In addition, the inflection point age of each trait was predicted to be between 4 and 10 weeks. Furthermore, this study revealed that body size traits were closely related to carcass traits in Lindian chickens. In summary, Lindian chickens are in a rapid growth stage before the age of 10 weeks, and better slaughter performance can be achieved through good feeding management during this stage. The reproductive traits and muscles are the main developmental focus after the age of 19 weeks, so it is important to adequately meet their energy requirements for subsequent good breeding performance.

On the fragility of the basis on the Hamilton–Jacobi–Bellman equation in economic dynamics

In this paper, we provide an example of the optimal growth model in which there exist infinitely many solutions to the Hamilton–Jacobi–Bellman equation but the value function does not satisfy this equation. We consider the cause of this phenomenon, and find that the lack of a solution to the original problem is crucial. We show that under several conditions, there exists a solution to the original problem if and only if the value function solves the Hamilton–Jacobi–Bellman equation. Moreover, in this case, the value function is the unique nondecreasing concave solution to the Hamilton–Jacobi–Bellman equation. We also show that without our conditions, this uniqueness result does not hold.

When is frugality optimal?

We consider an optimal growth model in which the decision maker chooses the consumption path that maximizes her utility over a finite planning horizon. At each instant of time, she can opt for a regular product that pollutes the environment, or a non-polluting frugal product, which procures, however, lower utility. Depending on the parameter values, different optimal paths can materialize, including some where a frugal lifestyle is on the rise, and others where polluting persists during the planning horizon.

Solution to an open question about optimal economic growth models

We prove that if the total factor productivity A of an aggregative economy is right at the barrier , with σ being the growth rate of labor force and λ the real interest rate, then the unique policy to optimally control the economy is the same as the one for optimally controlling weak economies, where . This result gives a complete answer for the interesting open question raised by Vu Thi Huong in her recent paper [Optimal economic growth problems with high values of total factor productivity. Appl Anal. 2022;101:1315–1329].

Optimal Growths from Two Perspectives, with Conditional Macro- and Micro-consistency: Dynamic Summation of Economic Variables Under Differential Regional Representative Production Functions

AbstractAlthough the representative theory of consumer/firm is widely used in particular forms of utility and production functions, is the optimal growth model adequately supported by a multi-regional microeconomic base in a “general form of utility and production functions”? Is the “optimization of macro-aggregate functions” consistent with the “summation of micro-optimal outcomes”? Economists have explored the problem of summation of macro-variables in a special form of production and utility function setting to find some theoretical foundations of macroeconomics in the case of heterogeneous representative consumers. However, heterogeneous representative producers have not been studied enough, and the conclusions are too dependent on the functional form. This study searches for the conditions for macroeconomics to reach micro-foundations in the framework of heterogeneous producers and general functions. Linking the entire macroeconomy through inter-regional capital flows, we distinguish between using and equity capital, construct a multi-regional dynamic optimal growth model in a homogeneous representative consumer and heterogeneous representative manufacturers, compare the summation of the regional dynamics with the aggregate macro-dynamics, and find that (A) the regional and macro-perspectives are generally inconsistent in their optimization results; (B) if the population growth rate is the same across regions, the macro-steady state is a weighted summation of the steady states; and (C) if the inverse of the absolute risk aversion is linearly additive to consumption, the model is reliable. In particular, CARA and CRRA-type utility functions are eligible. The eligible utility function can only be an exponential utility function or a power function utility function (the logarithmic utility function can be considered a degenerate power function utility).

The rate of discount on public investments with future bias in an altruistic overlapping generations model

A typical approach to evaluating public investment projects requires an appropriate rate for discounting future benefits and costs. In many developed countries, the practical discount rate for benefit–cost analysis has been calculated using the Ramsey formula in an optimal growth model, stretched back to the pioneering work presented by Frank Ramsey in 1928. Intergenerational altruism that is widely observed in our society generates future bias. Therefore, elected governments under democracies are future-biased. In such cases, the Ramsey formula is no longer valid. This paper derives a modified Ramsey formula for an appropriate social discount rate to be applied instead of the conventional Ramsey formula. We also examine properties of equilibrium policy. The degree of future bias depends on the political power of young people relative to elderly people. Demographic effects such as population aging negatively affect public investment through decreasing future bias effects. This paper contributes to the optimal theory of public investment for calculating an appropriate social discount rate, both theoretically and practically.

The rationality of adaptive decision-making and the feasibility of optimal growth planning

Rationality, the premise of economics, is an ideal behavioral norm. In the real world, however, intertemporal decision-making is based on adaptive behavioral principles from companies to individual households. It bases on managerial accounting procedures, whereby action plans are formulated and implemented, differences from actual results are recognized, and revisions accumulate over time. We take the intertemporal decision-making problem of households’ consumption/saving (investment) planning in this paper. And we compare the validity of rationality and adaptability as decision-making principles. First, rational decision-making in the optimal growth model leads to a unique path. However, optimal growth planning is practically unstable on the saddle-point path and can only realize if it assumes rationality leading to perfect foresight. On the other hand, the growth paths guided by budget-controlled adaptive decision-making are diverse and distributed in the myriad around the optimal growth path. This redundancy creates stability in the management and operation of the plan. Because through the trial-and-error process of planning and actual comparison, we can implement a more advantageous plan while allowing for multiple next-best goals, including the optimal growth path. Moreover, the numerical results show that the sequentially adaptive consumption/investment planning is comparable to the optimal growth plan on a social welfare basis calculated by accumulating consumption utility and is practically manageable. For example, paths that exceed 0.9 as a ratio to the optimal growth plan are reachable from the initial planning stage at a ratio of 0.58. Based on the above results, we can now analyze intertemporal economic problems with this realistic, practical, and simple method, replacing dynamic optimization ones.

Stand Structure Management and Tree Diversity Conservation Based on Using Stand Factors: A Case Study in the Longwan National Nature Reserve

The management of stand structure and the protection of species diversity are crucial to forest ecosystem service functions. Changbai Mountain is one of the three major forest regions in Northeast China, and understanding the characteristics of stand structure and the allometric relationship between stand structure variables and the correlation between stand factors and species diversity is the basis for achieving effective forest management and ecological function improvement. In this study, the typical deciduous broad-leaved mixed forest and coniferous and broad-leaved mixed forest of the Longwan Nature Reserve in the Changbai Mountain were taken as the research objects, and the stand structure and diversity were investigated in detail. Allometric growth equations were established using the breast height diameter (DBH), tree height, and crown width of the main dominant species. Pearson correlation analysis was conducted on the stand structure and diversity of the forest community. The experimental results showed that young forests with small diameters and small individuals were the main part of the stands. Optimal allometric growth models of the main tree species (Acer mandshuricum, Ulmus davidiana var. japonica, Juglans mandshurica, Acer mono, and Tilia amurensis) were 0.807D0.646, 0608D0.381H0.390, 0.502D0.533H0.295, 0.795D0.540H0.157, and 0.541D0.484H0.301, respectively, with R2 values ranging between 0.6 and 0.8, indicating a good fit for the models. Furthermore, tree density, crown width, and DBH were the main factors affecting tree diversity. The research results will provide theoretical support for the efficient management of forest stand structure and diversity conservation in the Longwan Reserve.

Analytical approaches for growth models in economics

This study deals with the group-theoretical analysis of nonlinear optimal control problems called the optimal growth model with the environmental asset and capitalist decision model of endogenous growth, which are expressed in terms of the current and present value Hamiltonian functions. The first-order conditions for optimal control problems based on Pontryagin’s maximum principle are considered. In addition, Lie point symmetries and their relation with the Prelle-Singer, λ-symmetries, adjoint symmetries, Darboux polynomials, and Jacobi’s last multiplier approaches are studied by analyzing the coupled nonlinear first-order ordinary differential equations corresponding to the first-order conditions. For both current and present Hamiltonian cases, the first integrals and the corresponding analytical (closed-form) solutions and graphical representations for the optimal control problems are represented.

ТЕХНОЛОГІЯ ВИКОРИСТАННЯ ТЕОРІЇ ОПТИМАЛЬНОГО КЕРУВАННЯ ДЛЯ МОДЕЛЮВАННЯ ОДНОПРОДУКТОВОЇ МАКРОЕКОНОМІКИ ЗРОСТАННЯ

Research in macroeconomic growth remains extremely relevant and important in the modern world. Macroeconomic growth focuses on analysing the dynamics of production, income, and consumption, enabling the development of scenarios for long-term development. The technology of using the theory of optimal control for modelling single-product growth macroeconomics plays an important role in understanding and forecasting economic development. The article proposes a technology for using the theory of optimal control to model a single-product growth macroeconomy. Technological tasks for modelling single-product macroeconomic growth are formulated, which can be initial (at the beginning of the production process), intermediate throughout the entire duration of the process, and final at the end state of production. In order to construct a model of optimal macroeconomic growth, several assumptions are taken into account. Specifically, gross output is decomposed into final output and production consumption. Production consumption is assumed to be proportional to gross output. Final output is further divided into non-production consumption, total investment, government expenditure, taxation, pollution abatement, and balance of trade (exports minus imports). To build the optimal process, where consumption rate acts as the control variable and specific capital represents the phase trajectory, a target function (goal criterion) is formalized. The target function aims to maximize the average (integral) discounted consumption over a specified time interval. Sufficient optimality conditions are used to investigate the model of optimal control. The Lagrange method is employed to account for the constraint on minimum consumption. This allows transforming the constrained optimization problem into an unconstrained optimization problem involving the optimization of two functions with multiple variables. During the investigation of the single-product macroeconomic growth model, an algorithm has been developed to calculate the optimal process for three formulated technological tasks under selected three-tier regimes. Among the built optimal processes, it is possible to select a priority optimal process.

A Marxian Optimal Growth Model for Economies with Minimum Subsistence Wages

This study constructs a two-sector two-class economic growth model to analyze an economy described in Marx’s Capital, where only the capitalist owns the means of production and maximizes the surplus value, while the worker provides labor in exchange for the minimum subsistence wage. Unlimited labor supply is the critical factor for the wage to be at the minimum subsistence level. In contrast to the Marxian optimal growth model, which indicates that the growth path in the capitalist economy follows a stable path to a steady state and is appropriate for analyzing the developed capitalist economy, our model demonstrates that, except in some rare cases, there is no stable path to a steady state in the economy and it is valid for analyzing an economy with an excessive labor supply. The results further indicate that it is common for capital to follow the process of unlimited self-growth, which causes the capitalist economy to be unstable.

Spatial growth theory: Optimality and spatial heterogeneity

Spatiotemporal dynamics are introduced in a standard Ramsey model of optimal growth in which capital moves toward locations where the marginal productivity of capital is relatively higher and initiate a study of the effects of nonlinear capital transport in terms of a linearized analysis around steady state solution solutions.The potential spatial heterogeneity of optimal growth, as seen from the point of view of an optimizing social planner, is examined. Our results suggest that for a high utility discount rate, the spatial capital flows induce the emergence of optimal spatial patterns, while for a low utility discount, a flat earth steady state is socially optimal. Furthermore, when spatial heterogeneities exist due to total factor productivity differences across locations, we identify conditions under which the spatial capital flows could intensify or weaken spatial inequalities.

Equilibrium in a Symmetric Game of Resource Extraction with Coalitional Structure

The game of resource extraction / capital accumulation is a stochastic nonzero-sum infinite horizon game, obtained as an extension of the well-known optimal growth model to m strategically competing players, who jointly posses a renewable resource. The existence of a Nash equilibrium in different, often symmetric, frameworks of the game received a significant attention in the scientific literature on the topic. The focus of this paper is to introduce the coalitional component to the symmetric problem. Specifically, we examine whether the game with a fixed coalitional structure admits stability against profitable coalitional deviations.It is assumed that the set of all players is partitioned into coalitions which do not intersect and remain consistent throughout the game. The members of each coalition are able to coordinate their actions and perform joint deviations in a cooperative manner. Such setting incorporates a natural concept of established social ties, which may reflect a potential context appearing in practical applications. The corresponding notion of equilibrium in the paper is expressed as a position, from which none of the set coalitions can deviate in a manner to increase a total reward of its members. Its existence is studied in the context of a certain symmetric resource extraction game model with unbounded utilities of the players. This model was studied in [12; 13], concluding a Stationary Markov Perfect Equilibrium existence in both symmetric and non-symmetric game structure. The first feature of the model is that the preferences of the players are considered to be isoelastic in the form of strictly concave power functions. Furthermore, the law of motion between states is set to follow a geometric random walk in relation to players' joint investments. We prove that the game within the formulated settings admits stability against profitable coalitional deviations for any partition on the set of agents. The method provides an algorithm for building the corresponding stationary strategies, which can be useful for practical purposes. Finally, we use two examples with different numerical configurations to illustrate possible patterns of how the individual rewards of the players vary depending on a coalitional structure, which is set at the beginning of the game.

On Simple Rules for the Social Cost of Carbon

The objective of this paper is to assess the use of simple rules for the social cost of carbon. It is shown that several interrelated objections may apply. The main issues are the following. First, the underlying theoretical models typically assume that the economy finds itself on a balanced growth path, implying that addressing the issue of designing policies for the short run, which play a role in the actual policy debate, are neglected. Second, for some cases the assumptions made regarding the marginal damages of high temperature or of arge atmospheric CO_{2} stocks are shown to be incompatible with other assumptions made. Third, typically the rules follow from an optimal growth model and associate the social cost for a particular year with GDP for that year, but it is not always acknowledged that it should be optimal rather than actual GDP for that year. I also go into the performance of simple rules as compared to first-best.

Regime switching optimal growth model with risk sensitive preferences

We consider a risk-sensitive optimization of consumption-utility on an infinite time horizon where the one-period investment gain depends on an underlying economic state whose evolution over time is assumed to be described by a discrete-time, finite-state, Markov chain. We suppose that the production function also depends on a sequence of independent and identically distributed (i.i.d.) random shocks. For the sake of generality, the utility and the production functions are allowed to be unbounded from above. Under the Markov regime-switching model, it is shown that the value function of optimization problem satisfies an optimality equation and that the optimality equation has a unique solution in a particular class of functions. Furthermore, we show that an optimal policy exists in the class of stationary policies. We also derive the Euler equation of optimal consumption. Furthermore, the existence of a joint stationary distribution of the optimal growth process and the underlying regime process is examined. Finally, we present a numerical solution by considering a power utility and some hypothetical values of parameters in a regime switching extension of the Cobb–Douglas production rate function.

Development of Diameter Growth Models by Thinning Intensity of Planted Quercus glauca Thunb. Stands

Background and objective This study was conducted to develop diameter growth models for thinned Quercus glauca Thunb. (QGT) stands to inform production goals for treatment and provide the information necessary for the systematic management of this stands. Methods This study was conducted on QGT stands, of which initial thinning was completed in 2013 to develop a treatment system. To analyze the tree growth and trait response for each thinning treatment, forestry surveys were conducted in 2014 and 2021, and a one-way analysis of variance (ANOVA) was executed. In addition, non-linear least squares regression of the PROC NLIN procedure was used to develop an optimal diameter growth model. Results Based on growth and trait analyses, the height and height-to-diameter (H/D) ratio were not different according to treatment plot (p > .05). For the diameter of basal height (DBH), the heavy thinning (HT) treatment plot was significantly larger than the control plot (p < .05). As a result of the development of diameter growth models by treatment plot, the mean squared error (MSE) of the Gompertz polymorphic equation (control: 2.2381, light thinning: 0.8478, and heavy thinning: 0.8679) was the lowest in all treatment plots, and the Shapiro-Wilk statistic was found to follow a normal distribution (p > .95), so it was selected as an equation fit for the diameter growth model. Conclusion The findings of this study provide basic data for the systematic management of Quercus glauca Thunb. stands. It is necessary to construct permanent sample plots (PSP) that consider stand status, location conditions, and climatic environments.

Analyzing a Maximum Principle for Finite Horizon State Constrained Problems via Parametric Examples. Part 2: Problems with Bilateral State Constraints

In this article, a maximum principle for finite horizon state-constrained problems is analyzed via parametric examples. These parametric examples resemble typical optimal growth problems in mathematical economics. Since the maximum principle is only a necessary condition for local optimal processes, a large amount of additional investigations are needed to obtain a comprehensive synthesis of finitely many processes suspected for being local minimizers. Our analysis not only helps to understand the principle in depth, but also serves as a sample of applying it to meaningful prototypes of optimal economic growth models. Problems with unilateral state constraints have been studied in Part 1 of the article. Problems with bilateral state constraints are addressed in this Part 2.

Selection of optimal software reliability growth models using an integrated entropy–Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) approach

A large number of software reliability growth models (SRGMs) have been studied to estimate the reliability of software systems over the past 40 years. Different models have been developed upon different sets of assumptions. A few models were developed in a practical environment by considering testing effort, testing coverage, time delay fault correction, and fault reduction factor. Generally, SRGMs are not dataset independent, and thus the selection of an appropriate SRGM for use in a specific application is a challenging task in the software reliability area. The wrong selection of SRGMs might create a wrong estimation of reliability and consequently a delay in the release of the software. To overcome this problem, we have proposed a unique hybrid entropy weight‐based multi‐criteria decision‐making (MCDM) method and Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) approach for the selection of a suitable SRGM and applied it for optimal selection and ranking of SRGMs. The proposed hybrid approach identifies the need of the relative importance of criteria for a given application without which inter‐criterion comparison cannot be accomplished. It requires a set of model selection criteria along with a set of SRGMs and their level of criteria for optimal selection. It successfully displays the result in terms of a merit value which is used to rank the SRGMs. The proposed approach has been validated on two data sets of the real software failure. The results of the study play a vital role for the decision maker to judge the suitability of the SRGM.

Asymmetries of financial openness in an optimal growth model

This paper analyzes international borrowing and lending in an optimal growth model with adjustment costs for investment, under both pre-commitment and lack of pre-commitment to debt repayment. We study the dynamics of the current account in the transition towards the balanced growth path, and derive the implications of financial openness for both the transition path and the balanced growth path. A comparison with financial autarky reveals that, under pre-commitment, and to the extent that countries start from different initial conditions, financial openness is beneficial for both poor and rich countries, as it allows them to engage in mutually beneficial inter-temporal trade. During the adjustment process, relatively poor countries experience higher consumption and investment compared to autarky, and thus experience current account deficits and accumulate net foreign debt. The inter-temporal tradeoff is asymmetric between poor and rich countries, in that poor countries experience lower steady state consumption, due to the need to service their accumulated foreign debt while the opposite happens in relatively rich countries. Under non pre-commitment, this asymmetric inter-temporal tradeoff results in a time inconsistency problem. Poor countries reach a point in the adjustment process after which it is welfare improving for them to default on their foreign debt. In the absence of pre-commitment mechanisms, international lenders anticipate these incentives, and international lending and borrowing breaks down. This time inconsistency problem can thus explain both the Feldstein-Horioka puzzle and the Lucas paradox that capital does not flow from rich to poor countries. Credible sanctions in the case of default and ceilings on international borrowing are analyzed as partial solutions to the time-inconsistency problem caused by this asymmetry between poor and ‘rich’ countries.