Maximal Paths in the von Neumann Model

  • Lionel W. McKenzie
Part of the International Economic Association Series book series (IEA)


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.


Utility Function Growth Theory Angular Distance Production Cone Price Vector 
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  1. Dorfman, R., Samuelson, P. A., and Solow, R. M. (DOSSO) (1958). Chap. 12 in Linear Programming and Economic Analysis, McGraw-Hill, New York.Google Scholar
  2. Fisher, F. M. (1962). ‘Properties of the von Neumann ray in decomposable and nearly decomposable technologies.’ Manuscript.Google Scholar
  3. Gale, D. (1956). ‘The closed linear model of production’, in Kuhn, H. W., and Tucker, A. W. (eds.), Linear Inequalities and Related Systems, Princeton University Press, pp. 285–303.Google Scholar
  4. Kemeny, J. G., Morgenstern, O., and Thompson, G. L. (KEMOTH) (1956). ‘A generalization of the von Neumann model of an expanding economy’, Econometrica, vol. 24, no. 2, April 1956, pp. 115–35.CrossRefGoogle Scholar
  5. McKenzie, L. (1963). ‘Turnpike theorems for a generalized Leontief model’, Econometrica, vol. 31, nos. 1–2, January–April 1963, pp. 165–80.CrossRefGoogle Scholar
  6. Nikaido, H. (1964). ‘Persistence of continual growth near the von Neumann ray’, Econometrica, vol. 32, nos. 1–2, January 1964, pp. 151–62.CrossRefGoogle Scholar
  7. Radner, R. (1961). ‘Prices and the turnpike, III. Paths of economic growth that are optimal with regard only to final states’, Review of Economic Studies, vol. XXVIII, no. 2, February 1961, pp. 98–104.CrossRefGoogle Scholar
  8. Thompson, G. L. (1956). ‘On the solution of a game-theoretic problem’, in Kuhn, H. W., and Tucker, A. W. (eds.), Linear Inequalities and Related Systems, Princeton University Press, pp. 275–84.Google Scholar
  9. Tucker, A. W. (1955). Game Theory and Planning, Oklahoma State University Press, Stillwater, Okla.Google Scholar
  10. von Neumann, J. (1937). ‘Über ein ökonomisches Gleichungs-System und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes’, in Menger, K. (ed.), Ergebnisse eines Mathematischen Kolloquiums, no. 8, 1937. Translated as ‘ A model of general equilibrium’, Review of Economic Studies, vol. XIII, no. 1, 1945–46, pp. 1–9.Google Scholar
  11. Winter, S. G., Jr. (1961). ‘A boundedness property of the closed linear model of production’, The Rand Corporation, P-2384, July 1961.Google Scholar

Copyright information

© International Economic Association 1967

Authors and Affiliations

  • Lionel W. McKenzie
    • 1
  1. 1.University of RochesterRochesterUSA

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