Lionel W. McKenzie

Optimal economic growth, turnpike theorems and comparative dynamics

Publisher Summary This chapter is concerned with the long-term tendencies of paths of capital accumulation that maximize, in some sense, a utility sum for society over an unbounded time span. However, the structure of the problem is characteristic of all economizing over time whether on the social scale or the scale of the individual or the firm. The mathematical methods that are used are closely allied to the old mathematical discipline, calculus of variations. The chapter discusses that the utility function depends on time, as in the standard theory of the calculus of variations. Also the function to be maximized is the sum of utility functions for each period over the future. It is described as a separable utility function over the sequence of future capital stocks and corresponds to the integral of calculus of variations. As the consumption of one period influences the utility of later consumption, the separability assumption is not exact. The treatment of utility in a period as dependent on initial and terminal stocks is not a restriction because the usual assumptions that make utility depend on consumption and consumption on production and terminal stocks implies that an equivalent utility depending on capital stocks exists. The chapter also discusses that the primary sources of the optimal growth model are aggregate savings programs and capital accumulation programs for an economy, the theorems, and methods of the subject find applications in other areas with increasing frequency.

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 ϱ 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 ϱ is near 1, a unique optimal stationary stock when ϱ is equal to 1, and the absence of cyclic paths on the von Neumann facet when ϱ is equal to 1. The results are illustrated in generalizing a model provided by Weitzman and Samuelson.

STABILITY OF EQUILIBRIUM AND THE VALUE OF POSITIVE EXCESS DEMAND

I SHALL PROVE two theorems using a new method in the problem of stability of equilibrium based upon the second method of Liapounov [4, p. 256ff.]. The novelty of method lies in the selection of the function V(p) whose decrease with time leads to the equilibrium position.2 This is the price weighted sum of the positive excess demands. I shall first prove the existence and stability3 in the large of the set of equilibrium points in the case of cross-elasticities which are nonnegative. The set of equilibrium points is compact and convex, and if the gross substitution matrix is indecomposable at equilibrium, the equilibrium is unique. When a numeraire is not present, it is possible to proceed beyond the limitation of nonnegative cross-elasticities to consider cases where certain weighted sums of the partial derivatives of excess demands with respect to prices are positive. This appears to be a natural generalization. Although the second theorem is primarily of local interest, one hardly need apologize for that. Global stability is not to be expected in general. This type of study was initiated by Walras [8, p. 170) and given its present formulation by Samuelson [5, p. 269]. I shall not elaborate on its limitations. Suffice it to say that, strictly interpreted, the groping for equilibrium which

The Dorfman-Samuelson-Solow Turnpike Theorem

IN THE PAST TWO YEARS several global turnpike theorems have been established, for the Ricardian or simple Leontief model by Morishima [5] and the author [3], for a von Neumann model with strict convexity near the turnpike by Radner [6], and for a generalized Leontief model with capital goods by the author [4]. However, the original theorem of Dorfman, Samuelson, and Solow [1] dealt with a neo-classical transformation function. This transformation function allows processes to be present in which there is joint production of current output, unlike the authors local theorem for the generalized Leontief model. On the other hand, the conclusions for local behavior are stronger than those of the global theorems of Radner and the author. Also the strict convexity assumption of Radner is not needed. The original theorem, however, was given for the case of only two goods, where the argument does not expose all the difficulties; and in the continuous model of Samuelson [7], which is stated for n goods, strict convexity is assumed. Moreover, the original proofs of the theorem do not proceed beyond the derivation of reciprocity of roots for the linear approximation to the efficiency conditions. It will be the purpose of my paper to give the original theorem (slightly weakened) a complete proof, one that does not require strict convexity and applies to any finite number of goods.

The Existence of Competitive Equilibrium over an Infinite Horizon with Production and General Consumption Sets

Although many theorems have been proved on the existence of competitive equilibrium in production economies with an infinite set of goods and a finite set of consumers, nearly all suff er from a major defect. The consumption possibility sets are required t o equal the positive orthant. This rules out trade in personal service s and it does not allow for substitutions between goods on the subsistence boundary. Using methods similar to B. Peleg and M. E. Yaari (1970), the authors show both equilibrium existence and core equivalence for economies with production and general consumption sets. Copyright 1993 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Maximal Paths in the von Neumann Model

I shall concern myself with the problem of optimal accumulation in the von Neumann model as it was initially posed by Dorfman, Samuelson, and Solow (1958) (DOSSO).2 In this problem the objective is to reach a point on a prescribed ray through the origin which is as far out as possible in a given number of periods. Let the prescribed ray be \( \left( {\bar y} \right) \). Then, if there is free disposal, and accumulation occurs over N periods from y 0 as a starting-point, it is equivalent to maximize the minimum of \( \frac{{y_i^T}}{{{{\bar y}_i}}} \) over i such that \( {\bar y_i} > 0 \). We may define \( \rho (y) = \min \frac{{{y_i}}}{{{{\bar y}_i}}}\,for\,{\bar y_i} > 0 \). Then ρ (y)is a utility function which is maximized.

Equilibrium, Trade, and Capital Accumulation

The paper summarizes the authors 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

Capital Accumulation Optimal in the Final State

The turnpike theorem for paths of accumulation in the von Neumann model which optimize final stocks is extended to general convex technologies. It is shown that optimal paths converge to a facet of the production cone, the von Neumann facet, on which there lies a path of maximal balanced growth for certain relevant goods. Then the asymptotic results are extended to a subset of the facet, which in many cases will be the path of maximal balanced growth itself, the von Neumann ray.

A note on comparative statics and dynamics of stationary states

Abstract Certain results are communicated regarding effects of parameter changes, when the stationary state is stable, on the stationary state and an optimal paths in its neighbourhood in discrete time multi-sectoral optimal growth models where period utilities are discounted.

A limit theorem on the core

Abstract The classical theorem of Debreu and Scarf (1963) on the relation of the core of an economy to its competitive equilibrium does not depend on an assumption that the preferences of consumers are convex, much less that they are strictly convex.