The hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth

It is shown that under very general circumstances, the standard optimal growth model with two or more capital goods can give rise to optimal trajectories that are limit cycles. An example with a nonjoint production Cobb-Douglass technology giving rise to closed cycles around a unique steady state is constructed. The stability of orbits is also studied.     

Saddle point property and Hopf bifurcation in continuous optimal growth models: a Lagrangian approach

Conditions for saddle point property, and the loss of it, have been widely studied. Generally these properties are established by means of a Hamiltonian formalism; we propose here to work without reference to any Hamiltonian system, and to use only the Lagrangian.Our study is local; it may seem that no new result can be obtained in this setting; nevertheless we establish sufficient conditions for the loss of saddle point property and for the existence of periodic orbits which, to our knowledge, are not found in the literature.We take the standard assumption that the Lagrangian is concave. It is well known that the cross derivatives of the Hamiltonian (i.e. H_xp(x, p)) are important in these problems, but the concavity-convexity property of the Hamiltonian does not easily give any information on these derivatives. On the other hand, we obtain such information directly in the Lagrangian version, because the Lagrangian is concave on its two arguments.We give here a self-contained version of our results and we do not hesitate to re-establish some well-known results, because we believe it is interesting to underline the straightforward aspect of the Lagrangian approach.     

Existence of optimal strategies in linear multisector models

In this paper, we give a sufficient and almost necessary condition for the existence of optimal strategies in linear multisector models when time is continuous, consumption is limited to one commodity, the instantaneous utility is of the CES type, and available technology allows a positive growth rate.JEL Classification Numbers: C62, O41 We thank an anonymous referee of this journal for careful scrutiny and very useful suggestions     

Mapping prices into productivity in multisector growth models

Two issues related to mapping a multi-sector model into a reduced-form value-added model are often neglected: the composition of intermediate goods, and the distinction between the productivity indices for value added and for gross output. We illustrate their significance for growth accounting using the well known model of Greenwood et al. (in Am Econ Rev 87(3):342–362, 1997), who find that about 60% of economic growth can be attributed to investment-specific technical change (ISTC). We investigate the role of intermediate goods in their framework and find that, taking into account the composition of intermediates, ISTC may well account for between 93 and 96% of post-war US growth.     

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.     

A discrete model of optimal economic growth

Contrary to most of the literature on optimal economic growth a discrete rather than a continuous model is investigated. It is shown that in such a discrete model it is easy to account for relatively freely changing functions and parameters. Thus, the production function, labor, the investment ratio, and the parameters for time preference, marginal utility, and depreciation are all allowed to depend on time. Using discrete dynamic programming methods, optimal investment policies are determined explicitly. These generalize important results from previous literature on optimal economic growth.     

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.      

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.     

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 chaos in optimal growth models with low discount rates

Summary We show that for every discount factor(0,1) one can find infinitely many strictly concave discrete-time optimal growth models in reduced form which have optimal policy functions exhibiting ergodic chaos. These reduced form models are interpreted in a two-sector optimal growth setting with utility functions depending on consumption as well as on capital.We thank Luigi Montrucchio for pointing out an error in a previous version of the paper.     

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     

A local analysis of stability and regularity of stationary states in discrete symmetric optimal capital accumulation models

In discrete time optimal capital accumulation models, where the period utilities have a certain “symmetry” property, a complete characterization of local stability of stationary states is obtained in terms of the characteristic roots of the product of certain submatrices of the Hessian of the utility function. This is used to analyze the connection between stability and properties of the effects of parameter changes on the stationary state. Certain differences in the results in discrete and continuous models are noted. A special case where utilities are “separable” in initial stocks and net investment is also treated.     

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.     

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.     

Equilibrium dynamics in discrete-time endogenous growth models with social constant returns

The existing literature establishes possibilities of local determinacy and dynamic indeterminacy in continuous-time two-sector models of endogenous growth with social constant returns. The necessary and sufficient condition for local determinacy is that the factor intensity rankings of the two sectors are consistent in the private/physical and social/value sense. The necessary and sufficient condition for dynamic indeterminacy is that the final (consumable) good sector is human (pure) capital intensive in the private sense but physical (consumable) capital intensive in the social sense. This paper re-examines the dynamic properties in a discrete-time endogenous growth framework and finds that conventional propositions obtained in continuous time need not be valid. It is shown that the established necessary and sufficient conditions on factor intensity rankings for local determinacy and dynamic indeterminacy are neither sufficient nor necessary, as the magnitudes of time preference and capital depreciation rates both play essential roles. We have benefitted from discussion with Robert Becker, Eric Bond, Michael Kaganovich, Karl Shell and participants of the Midwest Macroeconomic Conference in Chicago and the Midwest Economic Theory and International Trade Meetings at Indiana University. The fourth author acknowledges financial support from the Institute of Economics and Business Administration of Kobe University and the Institute of Economic Research of Kyoto University to enable this international collaboration.