On Sustained Growth Under Uncertainty

We analyze a stochastic one‐sector model of economic growth and investigate the conditions under which long‐run growth occurs almost surely. In contrast to the deterministic version of the model, the utility function plays a crucial role in determining the long‐run behavior of output.     

Human capital and the time-profile of human fertility

This is a 1st attempt at an explicitly intertemporal, microeconomic theory of the distribution of births over a woman's fertile period. The main results are that: 1) the optimal time-profile will satisfy the Hotelling rule for the depletion of a natural resource (in the present case of the women's stock of human capital at marriage); and 2) under certain simplifying assumptions, women with "high" initial endowments of human capital will have all their children in the 1st part of married life and then return to full-time employment, while women with "low" initial endowments will spread childbearing more evenly over the fertile period. It must also be pointed out that the characteristics of the optimal fertility profile might be different if utility depended directly on the profile itself--e.g., if the quality of children increased with the interval between the pregnancies, as assumed in the literature on birth spacing.     

Estimates of optimal public capital stocks in Japan using a public investment discount rate framework

The purpose of this paper is to empirically assess the optimality of the level of public capital in Japan. We use a methodological approach based on Burgess's (1988) procedure for calculating the public discount rate. This approach involves estimating a production function, but does not necessarily require utility function estimation. The results indicate that, although the Japanese economy experienced a public capital deficiency over the period 1960–1982, public capital moved toward optimal levels throughout the period. First version received: March 1997/final version received: June 1998     

A Golden Formula in Neoclassical‐Growth Models with Brownian‐Motion Shocks

In the paper, a Golden Formula, which does not depend on the specification of production and preference functions, is established to reveal that time‐average of the growth rate of optimal capital accumulation will converge to a constant, which is endogenously determined by relevant parameters, almost surely. The Golden Formula naturally implies surprisingly interesting and also intrinsic economic relations between some important macroeconomic variables; for example, it serves as a direct bridge between the modified Golden Rule and the modified Ramsey Rule. Furthermore, it indeed subsumes and hence substantially extends the classical Golden Rule in deterministic theory.     

Stochastic growth when utility depends on both consumption and the stock level

Summary This paper examines the dynamic behavior of optimal consumption and investment policies in the aggregate stochastic growth model when utility depends on both consumption and the stock level. Such models arise in the study of renewable resources, monetary growth, and growth with public capital. The paper shows that there is a global convergence of optimal policies to a unique stationary distribution if (a) there is sufficient complementarity in the model, or (b) if there is sufficient randomness in production. Two examples illustrate the possibility of multiple stationary distributions. In one, multiple stochastic steady states exist for a generic class of production and utility functions.We thank Professors Jess Benhabib and R. Robert Russell for helpful discussions and 2 referees for constructive suggestions.     

A Behavioral Learning Process in Games

This paper studies the cumulative proportional reinforcement (CPR) rule, according to which an agent plays, at each period, an action with a probability proportional to the cumulative utility that the agent has obtained with that action. The asymptotic properties of this learning process are examined for a decision-maker under risk, where it converges almost surely toward the expected utility maximizing action(s). The process is further considered in a two-player game; it converges with positive probability toward any strict pure Nash equilibrium and converges with zero probability toward some mixed equilibria (which are characterized). The CPR rule is compared in its principles with other reinforcement rules and with replicator dynamics. Journal of Economic Literature Classification Number: C72.     

Optimal fertility along the life cycle

We explore the optimal fertility timing in a four-period OLG economy with physical capital, whose specificity is to include not one, but two reproduction periods. It is shown that, for a given total fertility rate, the economy exhibits quite different dynamics, depending on the timing of births. If all births take place in the late reproduction period, there exists no stable stationary equilibrium and the economy exhibits cyclical dynamics due to labor growth fluctuations. We characterize the long-run social optimum and show that optimal consumptions and capital depend on the optimal cohort growth factor, so that there is no one-to-one substitutability between early and late fertility. We also extend Samuelson’s Serendipity Theorem to our economy and study the robustness of our results to: (1) endogenizing fertility timing, (2) assuming rational anticipations about factor prices, (3) adding a third reproduction period.     

Utility and production externalities, equilibrium efficiency and leisure specification

This paper analyzes the implications that the specification of the leisure activity has on the equilibrium efficiency in a two-sector endogenous growth model with human capital accumulation. We consider external effects of consumption and leisure in utility, and sector-specific externalities associated to physical and human capital in production. The optimal tax policy to correct for the distortions caused by the externalities is characterized under all the typical leisure specifications considered in the literature: home production, quality time and raw time. We show that the optimal policy depends markedly on the leisure specification.     

Some results on “an income fluctuation problem”

A consumer at each period, given the income available, y, has to decide how much to consume and save. If he consumes c ? 0 units he gets u(c) units of satisfaction or utility, and if x = y ? c ? 0 is the amount saved then the available income in the next period is rx + ω^k, where ω^k is a random variable, and r is an interest factor that is assumed to be known with certainty. Infinite time horizon problems are considered, and it is shown that if 0 < δr < 1, where 0 < δ < 1 is a discount factor, then the limiting policy is optimal. Questions about the behavior of the stock level, such as boundness, are considered, and an example is given that shows that the stock level might converge almost surely to infinity. Finally an economic explanation is given.     

Fair Accumulation under Risky Lifetime

Individuals save for their old days, but not all of them enjoy the old age. This paper characterizes the optimal capital accumulation in a two‐period OLG model where lifetime is risky and varies across individuals. We compare two long‐run social optima: (1) the average utilitarian optimum, where steady‐state average welfare is maximized; (2) the egalitarian optimum, where the welfare of the worst‐off at the steady‐state is maximized. It is shown that, under plausible conditions, the egalitarian optimum involves a higher capital and a lower fertility than the utilitarian optimum. Those inequalities hold also in a second‐best framework where survival conditions are exogenously linked to the capital level.     

Non-convex aggregate technology and optimal economic growth

This paper examines a model of optimal growth where the aggregation of two separate well behaved and concave production technologies exhibits a basic non-convexity. First, we consider the case of strictly concave utility function: when the discount rate is either low enough or high enough, there will be one steady state toward which the convergence of the optimal path is monotone and asymptotic. When the discount rate is in some intermediate range, we will find sufficient conditions for having either one equilibrium or multiple equilibria steady state. Depending to whether the initial capital per capita is located with respect to a critical value, we show that the optimal paths monotonically converge to one single appropriate equilibrium steady state. Second, we consider the case of linear utility and provide sufficient conditions to have either unique or two steady states when the discount rate is in some intermediate range. In this range, we give conditions under which the above critical value might not exist, and the economy attains one steady state in finite time, then stays at the other steady state afterward. P. Michel passed away when this research was completed. This paper is dedicated to his memory as a friend and colleague. N. M. Hung and C. Le Van thank the referee for vey helpful remarks and criticisms. They are grateful to Takashi Kamihigashi for very fruitful discussions. They also thank J.-F. Leclerc for editing the final version of this paper.     

Dynamic optimization with a nonsmooth, nonconvex technology: the case of a linear objective function

This paper studies a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous. The model also allows for a general form of irreversible investment. We show that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured. These conditions generalize those known for the case of S-shaped production functions. We also show that as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption.This paper is dedicated to Professor Mukul Majumdar on his 60th birthday. His research with various co-authors in the late 70s and the 80s pioneered innovative techniques for the analysis of nonconvex dynamic optimization models – both deterministic and stochastic. Roy considers himself particularly fortunate for having had the opportunity to learn economic theory and mathematical economics from Professor Majumdar. This paper has benefited from helpful comments and suggestions by an anonymous referee. Financial support from the 21st Century COE Program at GSE and RIEB, Kobe University, is gratefully acknowledged.     

Optimal Two Sector Growth Models with Three Factors

The paper is devoted to construction of optimal trajectories in the model, which balances growth trends of investments in capital and labor efficiency. The model is constructed within the framework of classical approaches of the growth theory. It is based on three production factors: capital, educated labor and useful work. GDP level is described by a production function of the Cobb–Douglas type. The utility function of the growth process is given by an integral consumption index discounted on the infinite horizon. The optimal control problem is posed to balance investments in capital and labor efficiency. The problem is solved on the basis of dynamic programming principles. A novelty of the solution consists in constructing nonlinear stabilizers constructed on the feedback principle, which leads the system from any current position to a steady state. Growth and decline trends of the simulated trajectories are studied for all components included in the model.     

The ramsey model revisited: The optimality of cyclical consumption and growth

This paper reconsiders the familiar Ramsey model of optimal saving. This model is modified with two respects: direct spillovers (positive or negative) of capital on utility and penalties for changes of consumption. It will be shown that these two modifications may substantially change the stability properties of an optimal programme if capital provides utility directly (in addition to indirectly through production). More precisely, complex strategies such as (stable) limit cycles or even unstable policies may be optimal. Adjustment costs must be sufficiently high in order to obtain stable limit cycles (or even exploding spirals). This is quite surprising because large adjustment costs are expected to smooth intertemporal strategies.     

Optimal stationary consumption with stochastic production and resources

Uncertainty about technology and resources is represented in terms of uncertainty about an (exogenous) environment whose successive states form a stationary stochastic process, with probabilities that are unaffected by economic decisions. The successive states of the economy depend both on the environment and on the decisions taken with regard to production and consumption. It is shown that, under conditions that are natural extensions of “neoclassical” conditions in the case of certainty, (1) Capital saturation is possible, i.e., an optimal stationary stochastic program exists, and (2) An optimal program can be sustained by a price system that takes the form of a stationary stochastic process of price vectors. In other words, an optimal stationary program can be sustained by a stochastic “equilibrium,” in which at each date the optimal production decisions maximize expected intertemporal profit, and the optimal aggregate consumption vector has minimum cost among all aggregate consumption vectors yielding no less (social) utility.     

Tolerance and Herd Behavior: Sophistication of Ideas under Payoff Complementarity

In this paper, I first show that there may be inefficient herd behavior even with a continuous choice set and a continuous loss function if there is payoff complementarity. Then, I show that the probability of inefficient herd behavior is asymptotically zero and the choice sequence converges to the optimal one almost surely if people have even a small amount of tolerance that they are willing to accept one another as a partner in joint work. This result is closely related to the argument in J. S. Mill’s On Liberty, where he states that tolerance for others’ ideas is essential for sophistication of ideas.     

Turnpike theory, discounted utility, and the von Neumann facet

The convergence of infinite optimal paths to stationary optimal paths is proved in models of capital accumulation whose discount factors are near 1, where strict concavity is not required for utility functions and production functions. The critical assumptions are unique support prices for points of the von Neumann facet, where is near 1, a unique optimal stationary stock when is equal to 1, and the absence of cyclic paths on the von Neumann facet when is equal to 1. The results are illustrated in generalizing a model provided by Weitzman and Samuelson.     

The taxation of capital returns in overlapping generations models

This paper shows that in the Diamond (1965) overlapping generations economy with production and capital savings, there is a period-by-period balanced fiscal policy supporting a steady state allocation that Pareto-improves upon the laissez-faire competitive equilibrium steady state (whether dynamically inefficient or efficient) without resorting to intergenerational transfers. The policy consists of taxing linearly (or subsidizing, in the dynamically efficient case) the returns to capital, while balancing the budget period by period through a lump-sum transfer (or tax, respectively) in second period. This intervention grants every generation the highest steady state utility attainable through markets (i.e. remunerating factors by their marginal productivities and without transfers) which under laissez-faire is not a competitive equilibrium outcome. A transition from the competitive equilibrium steady state to this other steady state is also Pareto-improving when the former is dynamically inefficient. The result disentangles from redistributive considerations the impact of the taxation of capital returns on steady state welfare, and thus provides a rationale for the taxation of capital returns that is based on efficiency considerations and not on redistributive goals.     

ON THE ROLE OF THE GROWTH OPTIMAL PORTFOLIO IN FINANCE

The paper discusses various roles that the growth optimal portfolio (GOP) plays in finance. For the case of a continuous market we show how the GOP can be interpreted as a fundamental building block in financial market modeling, portfolio optimisation, contingent claim pricing and risk measurement. On the basis of a portfolio selection theorem, optimal portfolios are derived. These allocate funds into the GOP and the savings account. A risk aversion coefficient is introduced, controlling the amount invested in the savings account, which allows to characterize portfolio strategies that maximise expected utilities. Natural conditions are formulated under which the GOP appears as the market portfolio. A derivation of the intertemporal capital asset pricing model is given without relying on Markovianity, equilibrium arguments or utility functions. Fair contingent claim pricing, with the GOP as numeraire portfolio, is shown to generalise risk neutral and actuarial pricing. Finally, the GOP is described in various ways as the best performing portfolio.     

The analytical solution of balanced growth of non-linear dynamic multi-sector economic model

In a one-sector neoclassical dynamic economic growth model, a reasonable ratio of investment to consumption exists, i.e., the “Golden Rule of Consumption”. This study is to extend one-sector neoclassical growth model to a multi-sector one. It is assumed that both the production function and the utility function are of Cobb–Douglas type, and the analytical expression of the balanced growth solution of the multi-sector model is provided, mainly including analytical expressions of the optimal distribution coefficient of fixed capital investment, the optimal distribution coefficient of labor hour, the proportion of production, the economic growth rate, the rate of change of the price index, and rental rates of different fixed capital.