Oscillations in optimal growth models

This paper discusses a model of optimal growth with non-zero discount rate. Most known results concern sufficient conditions for saddle-point (SP) property of the (unique) equilibrium point Here necessary and sufficient conditions for (local) SP are found which permits one to apply bifurcation theory. In particular, the paper considers bifurcation of periodic orbits from an equilibrium point by means of the Hopf theorem, thereby generalizing a result obtained by Benhabib and Nishimura for a special case. A nonconventional theory of the trade cycle may thus be based on very conventional assumptions. Finally, certain known (SP) stability conditions are discussed and related to the main result of the paper.     

The existence of optimal consumption policies in optimal economic growth models with nonconvex technologies

An existence theorem for a class of continuous time infinite horizon optimal growth models is developed. The underlying technology set is not assumed to be convex, instead the “slices” of the technology set corresponding to a fixed capital stock vector are assumed convex and compact in the consumption and net investment variables. This allows consideration of the case of increasing returns to scale. Existence of an optimal capital stock and consumption policy is proved directly without consideration of the underlying Hamiltonian dynamical system that arises from applying Pontryagin's maximum principle.     

Ergodic fluctuations in deterministic economic models

Nonperiodic business fluctuations are shown to exist with positive measure generically and to behave like stationary stochastic processes in the standard dynamic macro model. The implication is that so called ‘chaotic’ fluctuations are probable for random initial conditions in a large class of recursive economies.     

Existence of solutions in continuous-time optimal growth models

This paper studies the existence of solutions in continuous time optimization problems. It provides a theorem whose conditions can be easily checked in most models of the optimal growth theory, including those with increasing returns and multi-sector economies.      

Computation of non-linear continuous optimal growth models: experiments with optimal control algorithms and computer programs

In this paper we discuss the use of optimal control methods for computing non-linear continuous optimal growth models. We have discussed various recently developed algorithms for computing optimal control, involving step-function approximations, Runge–Kutta solutions of differential equations, and we suggest that the discretization approach is preferable to methods which solve first-order optimality conditions. We have surveyed some powerful computer programs by : , and for computing such models numerically. These programs have no substantial optimal growth modelling applications yet, although they have numerous engineering and scientific applications. A computer program named by is developed in this study. Results are reported for computing the Kendrick–Taylor optimal growth model using and programs based on the discretization approach. References are made to the computational experiments with and . The results are used to compare and evaluate mathematical and economic properties, and computing criteria. While several computer packages are available for optimal control problems, they are not always suitable for particular classes of control problems, including some economic growth models. The -based and , however, offer good opportunities for computing continuous optimal growth models. It is argued in this paper, that optimal growth modellers may find that these recently developed algorithms and computer programs are relatively preferable for a large variety of optimal growth modelling studies.     

Solving optimal growth models with vintage capital: The dynamic programming approach

This paper deals with an endogenous growth model with vintage capital and, more precisely, with the AK model proposed in [R. Boucekkine, O. Licandro, L.A. Puch, F. del Rio, Vintage capital and the dynamics of the AK model, J. Econ. Theory 120 (1) (2005) 39-72]. In endogenous growth models the introduction of vintage capital allows to explain some growth facts but strongly increases the mathematical difficulties. So far, in this approach, the model is studied by the Maximum Principle; here we develop the Dynamic Programming approach to the same problem by obtaining sharper results and we provide more insight about the economic implications of the model. We explicitly find the value function, the closed loop formula that relates capital and investment, the optimal consumption paths and the long run equilibrium. The short run fluctuations of capital and investment and the relations with the standard AK model are analyzed. Finally the applicability to other models is also discussed.     

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 $l_{\infty }^{+}$ . 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.     

Robust chaos in dynamic optimization models

The purpose of this paper is to investigate the (theoretical) importance of chaos as a phenomenon occurring in dynamic optimization problems. The intertemporal models we focus on are specified by a standard aggregative production function, an immediate return function depending on current consumption, capital input and a taste parameter, and a discount factor.We interpret “chaos” as a situation in which the Liapounov exponent of the relevant dynamical system is positive. This notion of chaos is related to the concept of “unpredictability” as measured by the Kolmogorov-Sinai entropy.In the family of intertemporal models, indexed by the taste parameter (with values lying in a closed interval), chaos is considered to be an “unimportant” phenomenon, if the set of parameter values for which chaos occurs is of Lebesgue measure zero.We identify a family of dynamic optimization models, for which the optimal transition functions are represented by the quadratic family of maps. Relying on the mathematical literature on the robustness of chaos for this family of maps, we conclude that chaos cannot be considered to be an unimportant phenomenon in dynamic optimization models.     

Hopf bifurcation and quasi-periodic dynamics in discrete multisector optimal growth models

This paper discusses the asymptotic stability of the steady state and the existence of a Hopf bifurcation in discrete time multisector optimal growth models. We obtain on the one hand a local turnpike theorem which guarantees the saddle point property for all discount rates. On the other hand, we provide a new proposition which gives some conditions ensuring local stability of the steady state if the impatience rate is not too high. A characterization of the boundδ*, above which the steady state is saddle-point stable, is also proposed in terms of indirect utility function's concavity properties. On this basis, some sufficient conditions for the existence of a Hopf bifurcation are stated. We thus prove the existence of quasi-periodic optimal paths in asymmetric models.     

On Lipschitz continuity of policy functions in continuous-time optimal growth models

Summary. This paper proves the C ^1,1 differentiability of the value function for continuous time concave dynamic optimization problems, under the assumption that the instantaneous utility is C ^1,1 and the initial segment of optimal solutions is interior. From this result, the Lipschitz dependence of optimal solutions on initial data and the Lipschitz continuity of the policy function are derived, by adding an assumption of strong concavity of the integrand. Received: July 29, 1996; revised version: November 25, 1997     

Stability of stochastic optimal growth models: a new approach

The paper proposes an Euler equation technique for analyzing the stability of differentiable stochastic programs. The main innovation is to use marginal reward directly as a Foster-Lyapunov function. This allows us to extend known stability results for stochastic optimal growth models, both weakening hypotheses and strengthening conclusions.     

The nature of the steady state in models of optimal growth under uncertainty

Summary. We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent . Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1-p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever . More delicate is the case . Singularity with respect to Lebesgue measure also appears for values such that . For and Peres and Solomyak (1998) have shown that the distribution is a.e. absolutely continuous. Characterization of the invariant distribution in the remaining cases is still an open question. The entire analysis is summarized through a bifurcation diagram, drawn in terms of pairs .Received: 9 April 2002, Revised: 29 October 2002, JEL Classification Numbers: C61, O41.Correspondence to: Tapan MitraThis research was partially supported by CNR (Italy) under the "Short-term mobility" program and by M.U.R.S.T. (Italy) National Group on "Nonlinear Dynamics and Stochastic Models in Economics and Finance" . We are indebted to Rabi Bhattacharya for providing us with the reference to Solomyak's (1995) paper. The present version has benefitted from comments by Mukul Majumdar and two anonymous referees.     

Robust ergodic chaos in discounted dynamic optimization models

Summary We identify a family of discounted dynamic optimization problems in which the immediate return function depends on current consumption, capital input and a taste parameter. The usual monotonicity and concavity assumptions on the return functions and the aggregative production function are verified. It is shown that the optimal transition functions are represented by the quadratic family, well-studied in the literature on chaotic dynamical systems. Hence, Jakobson's theorem can be used to throw light on the issues of robustness of ergodic chaos and sensitive dependence on initial conditions.     

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.     

On rational dynamic strategies in infinite horizon models where agents discount the future

In this paper we show that the equilibrium dynamics, arising in three models of rational choice over an infinite horizon, are described by choice functions belonging to huge classes of maps including chaotic ones The results are obtained by studying the relationship between the primal functions of the models and the corresponding choice functions. The three models are: the economic growth model with discounting, a duopoly game with alternate moves, a duopoly with simultaneous moves.     

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