Turnpike Theorem Research Articles

A turnpike theorem for continuous‐time control systems when the optimal stationary point is not unique

We study the turnpike property for the nonconvex optimal control problems described by the differential inclusion . We study the infinite horizon problem of maximizing the functional as T grows to infinity. The turnpike theorem is proved for the case when a turnpike set consists of several optimal stationary points.

Turnpike theorem for nonautonomous infinite dimensional discrete-time control systems

In this work we study the structure of “approximate” solutions for a nonautonomous infinite dimensional discrete-time control system determined by a sequence of con-tinuous functions vi : where X is a complete metric space. We show that for a generic sequence of functions there exists a sequence (the “turnpike”) such that the following properties hold: (i) is an optimal solution for any finite interval [k1k2 ]; (ii) given ϵ > 0, each “approximate” solution on an interval [k1 k2 ] with sufficiently large k 2-k 1 is within ϵ of the “turnpike” for all where L is a constant which depends only on ϵ

Stability and Turnpike Theorems in Dynamic Competitive Equilibrium

This study reviews the main results in the literature on the integration of competitive equilibrium theory and optimal growth theory, in particular those concerning the convergence of an equilibrium path to a stationary state (stability theorems). JEL Classification Numbers: C6, D9.

Equilibrium, Trade, and Capital Accumulation

The paper summarizes the author's principal contributions to economic theory: (1) one of the first rigorous proofs of the existence of competitive equilibrium; (2) existence of competitive equilibrium with weakened assumptions; (3) the minimum income approach to demand theory; (4) tatonnement stability with weak gross substitutes; (5) a general theory of comparative advantage; (6) factor price equalization with attention to factor supplies; and (7) turnpike theory allowing for von Neumann facets and neighbourhood convergence. JEL Classification Number: B10

A turnpike property of optimal programs for a class of simple linear models of production

This paper establishes a ‘turnpike theorem’ for a closed linear model of production with a primitive input requirement matrix. Optimal programs of resource allocation have a ‘turnpike property’ if the growth factor of every sector in the economy converges, in the long run, to a common value. The usefulness of such a theorem is due to the fact that the input requirement matrix for an economy with a large number of goods may be primitive (some power of the matrix is strictly positive).

Neighborhood Turnpike Theorem for Continuous-Time Optimization Models

A neighborhood turnpike theorem is proved for continuous-time, infinite-horizon optimization models with positive discounting. Our approach is a primal one and no differentiability assumption is ma...

STOCHASTIC VERSION OF POLTEROVICH'S MODEL: EXPONENTIAL TURNPIKE THEOREMS FOR EQUILIBRIUM PATHS

The paper examines a stochastic analogue of the Polterovich model of economic dynamics. The study focuses on the asymptotic properties of equilibrium paths. The main results are stochastic turnpike theorems with exponential estimates of the convergence rate.

Neighborhood Turnpike Theorem for Continuous-Time Optimization Models

A neighborhood turnpike theorem is proved for continuous-time, infinite-horizon optimization models with positive discounting. Our approach is a primal one and no differentiability assumption is made. The basic hypothesis is a condition of uniform concavity at the point defining the undiscounted steady state. The main novelty here is that we formulate the theorem by taking the undiscounted steady state as the turnpike.

Structure of optimal trajectories of convex processes

In this paper we study the dynamic properties of optimal trajectories of convex processes where K⊂ R nis a closed convex cone. We show that optimal trajectories of a convex process Gspend most of the time in a small neighborhood of a von Neumann path. The turnpike theorem that we obtain generalizes the result of Rubinov [10] which was established for a convex process where is the cone of the elements of the Euclidean space R nthat have nonnegative coordinates. Also, we show that the turnpike phenomenon is stable under small pertubations of a convex process G.

Consumption and portfolio turnpike theorems in a continuous-time finance model

Abstract This paper investigates consumption and portfolio turnpike theorems in a continuous-time model. When the inverse functions of the derivative of utility functions for consumption and investment belong to a special subclass of regularly varying functions, it is shown that optimum portfolio, final wealth and consumption processes for these utility functions can be approximated arbitrarily closely in a suitable sense by those for the corresponding power utility functions. As an immediate consequence, the consumption and investment turnpike theorem is established. Conversely, it is shown that the sufficient condition is also necessary for the turnpike property. Our results generalize those of Cox and Huang (1992) .

Lyapunov Sequences and a Turnpike Theorem without Convexity

A turnpike theorem is presented for a class of nonautonomous nonconvex difference inclusions defined by positively homogeneous increasing setvalued mappings. The proof involves a new concept of Lyapunov sequences based on Minkowski gauges of certain normal sets.

What Is Sustainable Development?

This paper introduces two axioms that capture the idea of sustainable development, and characterizes the welfare criterion that they imply. The axioms require that neither the present nor the future should play a dictatorial role in society's choices over time. Theorems 1 and 2 show there exist sustainable preferences which satisfy these axioms and provide a full characterization. Theorems 3-5 study a standard dynamical system representing the growth of a renewable-resource economy, give a turnpike theorem, and exhibit the differences between sustainable optima and the ones according to discounted utilitarianism.

Turnpike theorems in an integral dynamic model of economic renovation

Turnpike theorems in an integral dynamic model of economic renovation

Turnpike Theorems in Nonconvex Nonstationary Environments

This paper provides a comprehensive development of turnpike theory in a stochastic aggregative model with time-varying nonconvex technology. A new approach to turnpike theorems is developed that exploits the monotonicity of optimal programs and utilizes a supermartingale process generated by stochastic Euler equations. This extends the classical turnpike theory to general nonconvex, nonstationary environments. Copyright 1997 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Turnpike theorem for a class of differential inclusions arising in economic dynamics

In this work we study the asymptotic behavior of optimal trajectories for a class of differential inclusions of the form [xacute](t)$isinv;b(x(t)) - x(t) arising in economic dynamics where b belongs to a space of set-valued mappings. We establish the existence of an open everywhere dense subset E of the space of set-valued mappings such that each mapping from E has the turnpike property

A Turnpike Theory for Infinite-Horizon Open-Loop Competitive Processes

This paper deals with a class of dynamic discrete time open-loop games played over an infinite time horizon. The equilibrium concept is defined in the sense of overtaking optimal responses by the players to the program choices of the opponents. We extend to this dynamic game framework the results obtained by Rosen for concave static games. We prove existence, uniqueness, and asymptotic stability (also called the turnpike property) of overtaking equilibrium programs for a class of games satisfying a strong concavity assumption (strict diagonal concavity).

Turnpike theorems for convex problems

We prove turnpike theorems for systems described by differential inclusions with convex graphs.

Turnpike theorem for a class of set-valued mappings

In this work we study the asymptotic behavior of optimal trajectories for a class of superlinear mappings arising in economic dynamics. We establish the existence of an open everywhere dense subset F of the space of set-valued mappings such that each mapping fron F has the turnpike property.

Optimal Programs on Infinite Horizon 2

We consider the limit behavior, as $N \to \infty $, of the expression $\sum _{i = 0}^{N - 1} v(x_i ,x_{i + 1} )$ for programs $\{ x_i \} _{i = 0}^\infty $ in a compact metric space K, where v is a real-valued function defined on $K \times K$. We study the structure of $(v)$-weakly optimal programs and establish a turnpike theorem for a generic continuous function v. We also prove the existence of an almost periodic $(v)$-optimal program for a generic continuous function v.

A new turnpike theorem for discounted programs

In this paper we present new results on the local and global convergence property of solutions to an optimization model where the objective function is a discounted sum of stationary one-period utilities. The asymptotic local turnpike is given without differentiability assumptions but imposing some mild curvature restrictions on the utility function. This approach allows us to get easy estimates on the range of discount factors and the size of the neighborhood for which the asymptotic property occurs. The paper concludes by providing two global turnpike theorems. The first one is an asymptotic theorem derived from a result similar to Scheinkman's visit lemma. The second one turns out to be a restatement of McKenzie's neighborhood turnpike theorem.