Optimal Growth Model Research Articles

A method for socially evaluating the effects of long-run demographic paths on living standards

AbstractBACKGROUNDThe paper is motivated by the need for improved social evaluation of prospective demographic change in order to better inform policies that are designed to reduce the very long-run costs of population ageing and to achieve sustainable economic development.OBJECTIVEWhat is the very long-run social value of a given demographic path? What is the value of changes in mortality, immigration, fertility, and labour force participation? How important are shorter-term demographic changes relative to very long-term effects in determining the social value of the demographic path?METHODSA new simulation method is applied for socially evaluating demographic paths, by separating a demographic path into a stable population component and a transition path component. Sensitivity analyses are conducted with respect to demographic assumptions, labour force participation assumptions, and consumption needs by age, returns to scale, and intergenerational value judgements.RESULTSThe application to Australia shows the considerable social cost, in terms of the loss of discounted consumption per capita, of improvements in mortality and gains from higher immigration and increased participation. The effect of fertility, however, is very sensitive to assumptions about the age-specific consumption needs of the population and social value judgements about intergenerational equity.CONCLUSIONSOur method socially evaluates the very long-run implications of specified constant fertility, mortality, and migration, giving consideration to both the transition path and the ultimate stable state. Mortality improvement is costly and higher immigration is beneficial. The impact of higher fertility is sensitive to assumptions about consumption needs and intergenerational equity.(ProQuest: ... denotes formulae omitted.)1. IntroductionThis paper proposes a method for socially evaluating the trajectories of long-run population projections. Governments in developed countries have, in recent decades, become increasingly concerned about the future population ageing that is reflected in their population projections, and which has implications for national prosperity, government budgets3, and sustainable economic development. A range of public policies have been introduced to either slow down population ageing or to ameliorate its effects on national prosperity and government budgets. Policies to slow down ageing include pro-fertility policies such as child subsidies of various kinds (McDonald 2006; Gauthier 2007; Guest and Parr 2010; Parr and Guest 2011; Guest and Parr 2013) and pro-immigration policies (Malmberg 2006). Policies to boost both supply and demand for older workers (OECD 2006)4 are designed to reduce the national economic burden of ageing. Population is also seen as a mediating factor in sustainable economic development. The Australian Government, for example, produced a Sustainable Population Strategy in 2011 (Commonwealth of Australia 2011). Such strategies, however, typically sidestep any social evaluation of the nation's prospective demographic path, and rather focus on the implications of demographic paths for the 'needs' of the population in terms of infrastructure such as water, energy, transport and communication, of government services, education, and training, and of environmental amenity.It is the public policy attention to the economic effects of demographic change that motivates the social evaluation of alternative demographic paths proposed in this paper. The standard approach taken to such evaluations in the population economics literature is to embed demographic structure into an intertemporal macroeconomic model of optimal economic growth and then simulate the long-run effects of demographic change. The seminal study is Cutler et al. (1990), which has spawned a large literature5. A key feature of these models is optimising behaviour of either individual agents or a social planner. …

Global patterns of phytoplankton nutrient and light colimitation inferred from an optimality‐based model

AbstractThe widely used concept of constant ”Redfield” phytoplankton stoichiometry is often applied for estimating which nutrient limits phytoplankton growth in the surface ocean. Culture experiments, in contrast, show strong relations between growth conditions and cellular stoichiometry with often substantial deviations from Redfield stoichiometry. Here we investigate to what extent both views agree by analyzing remote sensing and in situ data with an optimality‐based model of nondiazotrophic phytoplankton growth in order to infer seasonally varying patterns of colimitation by light, nitrogen (N), and phosphorus (P) in the global ocean. Our combined model‐data analysis suggests strong N and N‐P colimitation in the tropical ocean, seasonal light, and N‐P colimitation in the Northern Hemisphere, and strong light limitation only during winter in the Southern Ocean. The eastern equatorial Pacific appears as the only ocean area that is essentially not limited by N, P, or light. Even though our optimality‐based approach specifically accounts for flexible stoichiometry, inferred patterns of N and P limitation are to some extent consistent with those obtained from an analysis of surface inorganic nutrients with respect to the Redfield N:P ratio. Iron is not part of our analysis, implying that we cannot accurately predict N cell quotas in high‐nutrient, low‐chlorophyll regions. Elsewhere, we do not expect a major effect of iron on the relative distribution of N, P, and light colimitation areas. The relative importance of N, P, and light in limiting phytoplankton growth diagnosed here by combining observations and an optimal growth model provides a useful constraint for models used to predict future marine biological production under changing environmental conditions. © 2014. American Geophysical Union. All Rights Reserved.

On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function

We consider a class of infinite-horizon optimal control problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of that kind the initial state is fixed, no constraints are imposed on the behavior of the admissible trajectories at infinity, and the objective functional is given by a discounted improper integral. Earlier, for such problems, S.M. Aseev and A.V. Kryazhimskiy in 2004–2007 and jointly with the author in 2012 developed a method of finite-horizon approximations and obtained variants of the Pontryagin maximum principle that guarantee normality of the problem and contain an explicit formula for the adjoint variable. In the present paper those results are extended to a more general situation where the instantaneous utility function need not be locally bounded from below. As an important illustrative example, we carry out a rigorous mathematical investigation of the transitional dynamics in the neoclassical model of optimal economic growth.

Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain

We show how to enhance the performance of a Smolyak method for solving dynamic economic models. First, we propose a more efficient implementation of the Smolyak method for interpolation, namely, we show how to avoid costly evaluations of repeated basis functions in the conventional Smolyak formula. Second, we extend the Smolyak method to include anisotropic constructions that allow us to target higher quality of approximation in some dimensions than in others. Third, we show how to effectively adapt the Smolyak hypercube to a solution domain of a given economic model. Finally, we argue that in large-scale economic applications, a solution algorithm based on Smolyak interpolation has substantially lower expense when it uses derivative-free fixed-point iteration instead of standard time iteration. In the context of one- and multi-agent optimal growth models, we find that the proposed modifications to the conventional Smolyak method lead to substantial increases in accuracy and speed.

Saddle-node bifurcations in an optimal growth model with preferences for wealth habit

This paper studies the dynamic implications of preferences for wealth habit in a one-sector optimal growth model. We show that the dynamics may encounter saddle-node bifurcations with respect to the parameters of the preferences: the relative weight of wealth in utility and the degree of wealth habit. We analytically provide the monotone comparative statics and the continuity of the critical capital stock with respect to the discount factor, the relative weight of wealth in utility and the degree of wealth habit.

Количественная оценка фенотипической пластичности в модели оптимального роста растений

Количественная оценка фенотипической пластичности в модели оптимального роста растений

Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence

We establish some elementary results on solutions to the Bellman equation without introducing any topological assumption. Under a small number of conditions, we show that the Bellman equation has a unique solution in a certain set, that this solution is the value function, and that the value function can be computed by value iteration with an appropriate initial condition. In addition, we show that the value function can be computed by the same procedure under alternative conditions. We apply our results to two optimal growth models: one with a discontinuous production function and the other with “roughly increasing” returns.

Secondary instability of roughness-induced transient growth

Calculations are performed to analyze the stability of steady roughness-induced transient growth to unsteady fluctuations. The basic states consist of an optimal transient growth model and a previously computed direct numerical simulation that fully resolves the three-dimensional roughness element. It is shown that sub-optimal transient growth produced by surface roughness is much more susceptible to destabilization than optimal perturbation predictions. The factors contributing to this behavior are analyzed in detail. Scaling properties from experiments allow computation of stability bounds for realistic surface roughness. These results are also used to explain the critical behavior seen in transition behind three-dimensional roughness elements.

Intergenerational equity with individual impatience in a model of optimal and sustainable growth

Among the ethical objections to intergenerational impartiality is the violation of consumer sovereignty given that individuals are impatient. We accommodate that concern by distinguishing intra- and inter-generational discounting in an OLG model suitable for analyzing sustainability issues. Under the assumption of constant elasticity of marginal felicity, the optimum trajectory of aggregate consumption is guided, via the Ramsey condition, by the intergenerational discount rate but not the personal discount rate. In an economy with produced capital and a renewable resource, intergenerational neutrality results in a sustained growth path, without the necessity of a sustainability constraint, even in the presence of intragenerational impatience. We also find that green net national product remains constant along the optimal approach path to golden rule consumption.

Discounting, Values, and Decisions

How do discount rates affect agents’ decisions and valuations? This paper provides a general method to analyze this question, allowing stochastic and managed cash flows, stochastic discount rates, and time inconsistency and including arbitrary learning and payoff or utility processes. We show that some of these features can lead to counterintuitive answers (e.g., “a more patient agent stops earlier”), but we also establish, under simple conditions, theorems yielding robust predictions concerning the impact of discount rates on control and stopping decisions and on valuations. We apply our theory to models of search, experimentation, bankruptcy, optimal growth, investment, and social learning.

On the convergence of optimal solutions in infinite horizon discrete time models

We analyse the dynamic properties of a one-sector optimal growth model and then extend this analysis to the optimal growth model in which each generation cares about its own parent generation, its own consumption and the utility of its successor.

Short-Term Credit: A Monetary Channel Linking Finance to Growth

This paper develops a mechanism that links the combined monetary and financial role of intermediaries to the division of labor and endogenous growth. The model builds on an augmented Ramsey Cass Koopmans (RCK) model of optimal growth. First, by relaxing the assumption that each agent buys and sells at the same time an endogenous cash-in-advance constraint is created. The cash constraint is not binding for agents who borrow from intermediaries at the start of a period and repay the debt at the end of the period. Thus intermediated short-term credit is a solution to the monetary friction. Second to address the division of labor the symmetric n-good n-type structure of Kiyotaki and Wright’s search model of money is nested into each period of the model. Because each type of agent is more productive when his production is specialized, relaxing the cash constraint leads to a division of labor. Finally the exogenous growth of the RCK model is reinterpreted as endogenous growth due to a process of learning-by-doing. We find that financial intermediaries by relaxing the cash constraint promote the division of labor which generates a process of endogenous growth.

Exponential discounting bias

We address intertemporal utility maximization under a general discount function that nests the exponential discounting and the quasi-hyperbolic discounting cases as particular specifications. Under the suggested framework, the representative agent adopts, at some initial date, an optimal behavior that shapes her consumption trajectory over time. This agent desires to take a constant discount rate to approach the optimization problem, but bounded rationality, under the form of a present bias, deviates the individual from the intended goal. As a result, decreasing impatience will end up dominating the agent’s behavior. The individual will not be aware of her own time inconsistency and, therefore, she will not revise her plans as time elapses, what makes the problem relatively simple to address from a computational point of view. The general discounting framework is used to approach a standard optimal growth model in discrete time. Transitional dynamics and stability properties of the corresponding dynamic setup are studied. An extension of the standard utility maximization model to the case of habit persistence is also considered.

OPTIMUM SAVINGS AND OPTIMAL GROWTH: THE CASS–MALINVAUD–KOOPMANS NEXUS

This paper surveys the contributions of David Cass, Tjalling Koopmans, and Edmond Malinvaud over the decades during which modern optimal growth theory was developed. By utilizing material ranging from dissertations, drafts, and working papers through conference presentations, discussions, and published papers, we show that both Malinvaud and Cass had significant impacts on the evolution of Koopman's thought, and the development of his part of what is known as “the Cass–Koopmans model.” Based on our findings, we conclude that the modern optimal growth model should include the contributions of Malinvaud, and be retitled “the Cass–Malinvaud–Koopmans” model accordingly.

Optimal Pollution Abatement Under ‘Sustainable’ and Other Social Time Preferences

This paper analyses optimal pollution abatement expenditure and the pricing of pollution under alternative social time preferences, including ‘sustainable’ preferences, defined as those that are consistent with two axioms of sustainable development introduced by Chichilnisky (Land Econ 73:467–491, 1997). These axioms state essentially that neither the welfare of present nor future generations ought to be favoured over the other in determining the socially optimal path of economic development. The method is to calibrate a modified Ramsey model of optimal global growth and saving, where pollution is generated by the global output of goods and services. Pollution in turn reduces output. The simulation results illustrate how sensitive the optimal pollution price and abatement expenditure can be over time to assumptions about the social time preference rate. They also show that ‘sustainable’ preferences impose a lower burden on future generations in terms of the pollution price and abatement expenditure. Hence there is a case for governments to make explicit their value judgments about intergenerational welfare, in the context of their notion of sustainable development, when setting target pollution abatement levels and pollution prices over time.

Strategic interaction and dynamics under endogenous time preference

This paper presents a strategic growth model with endogenous time preference. Due to the potential lack of concavity and the differentiability of the value functions associated with each agent’s problem, we employ the theory of monotone comparative statics and supermodular games based on order and monotonicity properties on lattices. In particular, we provide the sufficient conditions of supermodularity for dynamic games with open-loop strategies based on two fundamental elements: the ability to order elements in the strategy space of the agents and the strategic complementarity which implies upward sloping best responses. The supermodular game structure of the model lets us provide the existence and the monotonicity results on the greatest and the least equilibria. We sharpen these results by showing the differentiability of the value function and the uniqueness of the best response correspondences almost everywhere and show that the stationary state Nash equilibria tend to be symmetric. Finally, we numerically analyze to what extent the strategic complementarity inherent in agents’ strategies can alter the convergence results that could have emerged under a single agent optimal growth model. In particular, we show that the initially rich can pull the poor out of the poverty trap even when sustaining a higher level of steady state capital stock for itself.

A never-decisive and anonymous criterion for optimal growth models

We address in this paper the question of the existence of a Social Welfare Function that would be sustainable and would allow us to obtain solutions to optimal growth models. We define sustainability by two new axioms called Never-decisiveness of the present and Never-decisiveness of the future. We first show that a SWF which has Never-decisiveness properties cannot be defined on a ball of l+∞. We must (i) restrict to the set of utility streams for which the value of the SWF is finite and (ii) introduce additional assumptions in order to obtain the Never-decisiveness properties. Our main result in this paper is therefore to show that the undiscounted utilitarian criterion is an anonymous and never-decisive criterion for optimal growth models. We consider the set of utilities of consumptions which are generated by a specific technology, namely a technology with decreasing returns for high levels of capital, and restrict ourselves to good programs, i.e., any program for which intertemporal utility is well defined.

Beyond GDP: Modelling Labour Supply as a ‘Free Time’ Trade-Off in a Multiregional Optimal Growth Model

In this paper we develop the standard utility function of a Ramsey-type optimal growth model to account for a ‘market-time’ vs. ‘free-time’ trade-off. To do so, we introduce a free-time preference coefficient that measures the utility gained by deviating from a maximum labor supply defined as the combination of a 95% labor force participation rate for the 20 to 69 year-old population, and 3000 annual working hours (50 effective 60-hour weeks). We calibrate this free-time preference coefficient for 12 world regions on statistical and projected data from the United Nations, the International Labor Organization and the OECD. We illustrate a prospective use of this modelling development by comparing the consequences of convergence of the free-time preference coefficients of all world regions to the contrasted Western European vs. United States value. Over the 21st century, compared to a business-as-usual trajectory defined by maintained regional disparities in free time preference, convergence to US free time preference induces a 0.3% decrease in global discounted labor market time, but a 4.2% increase in discounted global GDP sustained by a 2.5% increase in primary energy consumption that translates into a 1.7% increase in cumulated CO2-equivalent emissions; convergence to Western European free time preference decreases labor market time by 13.8%, GDP by 11.7%, primary energy consumption by 10.7% and cumulated CO2-equivalent emissions by 9.1%.

DISCOUNT RATE DISTORTIONS AND THE RESOURCE CURSE

AbstractEmpirical evidence has suggested a “resource curse” exists, in which countries with abundant resources may have higher initial consumption but then grow more slowly. The effect appears to be dependent on a country's political structure. Theoretical models not typically accounted for historical exceptions, or have not shown the effect exists in a dynamic growth setting. We derive the resource curse effect in an optimal growth model augmented with a political process. The economy has a finite nonrenewable resource, and the government planner can choose to over‐extract natural resources relative to the efficient path by distorting the discount rate, but in so doing incurs political costs that depend on the presence of democracy. Government planners in non‐democratic countries usually have more autonomy in policymaking than those in democratic countries; therefore, the political cost is lower for non‐democratic countries. We show that the incentive for the planner to distort the extraction path is larger, the higher is the initial resource endowment. Consistent with empirical evidence, the distortion raises short‐term consumption but lowers the long‐term growth rate, and institutional differences create corner solutions that explain why some resource‐abundant countries avoid the curse. These results are robust to the inclusion of autonomous technological change.

Forward looking dynamics in spatial CGE modelling

This paper sets up a spatial dynamic CGE framework by combining the optimal growth model of saving and investment under adjustment costs and the spatial CGE model with Dixit–Stiglitz structure in the modern sector. Because of increasing product diversity on the dynamic equilibrium path, the model belongs to the category of semi-endogenous growth models. We overcome the difficulty of existing multiregional models to correctly approximate the infinite horizon equilibrium by employing a theoretically consistent terminal condition. The distinction of goods, factors, firms, and households by location, and the incorporation of trade costs in the model allow to study a variety of issues in regional and transport economics. We describe the model calibration and a tailor-made solution algorithm. The functionality is demonstrated using two illustrative examples.