Long-Run Optimal Economic Growth: Turnpike Theorems
Krane, Spencer D.
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My paper is divided into four sections. The first three sections each deal with a growth model, its maximum growth path, and a Turnpike Theorem associated with1 this path, Section One deals with the von Neumann input-output model and the 1961 Radner proof of the turnpike properties of the von Neumann ray. This was the first successful mathematically rigorous proof of a Turnpike Theorem. Section Two considers the Neoclassical growth model and two theorems associated with the Phelp's Golden-rule path as a turnpike. These theorems use calculus of variations to prove the turnpike conjectures. The optimal growth path is determined by maximizing the utility of per capita consumption. Section Three works with the Dorfman·Samuelson-Solow linear growht model. Maximum growth is shown to be determined by the same factors as in the von Neumann case. A proof for paths in the locality of the von Neumann ray is presented. This and the second Neo-classical proof are attributed to Paul Samuelson. They both show that the optimal growth path will travel in catenatry fashion about the turnpike. The properties of catenaries are used to prove the theorems. Section four concludes the paper with a few remarks concerning differences in the assumptions and proofs of the various models and theorems. The Appendix contains several calculations and a proof of a characteristic of catenary motion whose results were used in the paper but which were excluded from the body for lucidity.