The Turnpike Theorem of Morishima Get access Lionel W. McKenzie Lionel W. McKenzie Rochester Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 30, Issue 3, October 1963, Pages 169–176, https://doi.org/10.2307/2296317 Published: 01 October 1963

Journal Article Specialization in Production and the Production Possibility Locus Get access Lionel W. McKenzie Lionel W. McKenzie Durham, North Carolina Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 23, Issue 1, 1955, Pages 56–64, https://doi.org/10.2307/2296152 Published: 01 April 1955

Journal Article Specialisation and Efficiency in World Production Get access Lionel W. McKenzie Lionel W. McKenzie Durham, North Carolina Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 21, Issue 3, 1953, Pages 165–180, https://doi.org/10.2307/2295770 Published: 01 January 1953

This paper presents the classical theorem on the existence of equilibrium as it was proved in the 1950's with the various improvements that have been made since then.In particular, the elimination of the survival assumption and of the requirement of transitive preferences are carried through with a proof that uses a mapping of social demand.This approach favors intuitive understanding and generalization of the results.Finally, the role of the firm and the introduction of external economies are critically viewed. MYPURPOSE IS TO DISCUSS the present status of the classical theorem on existence of competitive equilibrium that was proved in various guises in the 1950's by Arrow and Debreu [1], Debreu [5, 6], Gale [8], Kuhn [14], McKenzie [17, 18, 19], and Nikaido [22].The earliest papers were those of Arrow and Debreu, and McKenzie, both of which were presented to the Econometric Society at its Chicago meeting in December, 1952.They were written independently.The paper of Nikaido was also written independently of the other papers but delayed in publication.The major predecessors of the papers of the fifties were the papers of Abraham Wald [31, 32] and John von Neumann [30], all of which were delivered to Karl Menger's Colloquium in Vienna during the 1930's.The paper of von Neumann was not concerned with competitive equilibrium in the classical sense but with a program of maximal balanced growth in a closed production model.However, he first used a fixed point theorem for an existence argument in economics and provided the generalization of the Brouwer theorem that was the major mathematical tool in the classical proofs.Wald achieved the first success with the general problem of the existence of a meaningful solution to the Walrasian system of equations.The proofs which were published used an assumption that later became known as the Weak Axiom of Revealed Preference.This axiom virtually reduces the set of consumers to one person, since it is equivalent to consistent choices under budget constraints.In a one consumer economy the existence of the equilibrium becomes a simple maximum problem and advanced methods are not needed.When many consumers with independent preference orders are present, it has been shown (Uzawa [29]) that fixed point methods are necessary.Wald also wrote a third paper whose main theorem was announced in a summary article [33], but which never reached

[Support prices are derived for weakly maximal paths in an optimal growth model which is time dependent but without uncertainty. The notion of "reachable" stocks and paths is defined and used to derive turnpike theorems by the value loss method. The proofs do not depend on the presence of optimal balanced paths nor on the usual transversality conditions. The theorems are extended to the classical model which has a non-trivial von Neumann facet.]

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