# A Stochastic Theory of the Generalized Cobb-Douglas Production Function

Conference paper

## Abstract

Factor substitution relations and returns to scale found in empirical studies of production functions are modeled in a way that is fully integrated with technological change and process innovation. The model is based on the Pareto boundary of random *n*-tuples, which is a rectangular hyperbola, asymptotic to the axes, whose parameter depends on the amount of sampling (search) activity. Conversion from probability fractiles to natural units of measurement of the factors give the Cobb-Douglas factor substitution relations. The influence of inputs on outputs is somewhat different because an upper bound on output cannot be assumed.

## Keywords

Production Function Natural Unit Rectangular Hyperbola Contract Production Progress Function## Preview

Unable to display preview. Download preview PDF.

## References

- Alchian, A. A. (1959), “Costs and Outputs,” M. Abramovitz
*et al*. (Eds.), The Allocation of Economic Resources, Stanford University Press, pp. 23–40.Google Scholar - Arrow, K. J. (1962) “The Economic Implications of Learning by Doing,”
*Review of Economic Studies**29*, pp. 155–173.CrossRefGoogle Scholar - Chenery, H. B. (1949), “Engineering Production Functions,”
*Quarterly Journal of Economics**63*, pp. 507–531.CrossRefGoogle Scholar - Cobb, C. W., and P. H. Douglas (1928), “A Theory of Production,”
*American Economic Review 18 (Supplement)*, pp. 139–165.Google Scholar - Grossman, E. R. F. W. (1959), “A Theory of the Acquisition of Speed Skill,”
*Ergonomics**2*, pp. 153–166.CrossRefGoogle Scholar - Evenson, R. E., and Y. Kislev (1975), “A Stochastic Model of Applied Research,”
*Journal of Political Economy*84, pp. 265–281.CrossRefGoogle Scholar - Ferguson, A. R. (1951), “An Airline Production Function” (Abstract),
*Econometrica**19*, pp. 57–58.Google Scholar - Fisher, R. A., and L. H. C. Tippett (1928), “Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample,”
*Proceedings of the Cambridge Philosophical Society*24, pp. 180–190.CrossRefGoogle Scholar - Fréchet, M. (1927), “Sur la Loi de Probabilité de l’Écart Maximum,”
*Annales de la Societe Polonaise de Mathematiques**6*, pp. 93–116.Google Scholar - Galambos, J. (1978),
*The Asymptotic Theory of Extreme Order Statistics*,Wiley.Google Scholar - Gnedenko, B. (1943), “Sur la Distribution Limite du Terme Maximum d’une Serie Aleatoire,”
*Annals of Mathematics*44, pp. 423–453.CrossRefGoogle Scholar - Gumbel, E. J. (1958),
*Statistics of Extremes*,Columbia University Press.Google Scholar - Hirsch, Werner Z. (1952), “Manufacturing Progress Functions,”
*Review of Economics and Statistics*34, pp. 143–155.CrossRefGoogle Scholar - Kamien, Morton I., and Nancy L. Schwartz (1982),
*Market Structure and Innovation*,Cambridge University Press.Google Scholar - Kurz, M., and A. S. Manne (1963), “Engineering Estimates of Capital-Labor Substitution in Metal Machinery,”
*American Economic Review*53, pp. 662–681.Google Scholar - Lele, P. T., and J. W. O’Leary (1972), “Applications of Production Functions in Management Decisions,”
*American Institute of Industrial Engineers Transactions*4, pp. 36–42.Google Scholar - Levy, F. K. (1965), “Adaptation in the Production Process,”
*Management Science 11(B)*, pp. 136–154.Google Scholar - Mansfield, Edwin (1965), “Rates of Return from Industrial Research and Development,”
*American Economic Review 55 (Supplement)*, pp. 310–322.Google Scholar - Misès, R. de (1936), “La Distribution de la Plus Grande de
*n*Valeurs,”*Revue Mathematíque de l’Union Interbalkanique*1, pp. 141–160.Google Scholar - Muth, John F. (1982), “Process Life Cycles and ExperienceCurves,” ORSA/TIMS National Meeting, San Diego, October 25–27. (Indiana University, School of Business, Discussion Paper #199, October 25, 1982.)Google Scholar
- Muth, John F. (1986a), “Search Strategies for Invention and Productivity Improvement,” ORSA/TIMS National Meeting, Atlanta, November 4–6Google Scholar
- Muth, John F. 1985.. (Indiana University, School of Business, Discussion Paper #306, April 8, 1986.)Google Scholar
- Muth, John F. (1986b), “Search Theory and the Manufacturing Progress Function,”
*Management Science**32*, pp. 948–962.CrossRefGoogle Scholar - Muth, John F. (1988), “A Model of Learning Curves with Restricted Random Sampling.” (Indiana University, School of Business, Discussion Paper #391, July 5, 1988.)Google Scholar
- Roberts, Peter C. (1983), “A Theory of the Learning Process,”
*Journal of the Operational Research Society*34, pp. 71–79.Google Scholar - Rosen, Sherwin (1972): “Learning by Experience as Joint Production,”
*Quarterly Journal of Economics**86*, pp. 579–594.CrossRefGoogle Scholar - Sahal, D. (1979), “A Theory of Progress Functions,”
*American Institute of Industrial Engineers Transactions**11*, pp. 23–29.Google Scholar - Smith, V. L. (1957), “Engineering Data and Statistical Techniques in Analysis of Production and Technological Change: Fuel Requirements in the Trucking Industry,”
*Econometrica**25*, pp. 281–301.CrossRefGoogle Scholar - Telser, Lester G. (1982), “A Theory of Innovation and Its Effects,”
*Bell Journal of Economics*13, pp. 69–92.CrossRefGoogle Scholar - Tippett, L. H. C. (1925), “On the Extreme Individuals and the Range of Samples Taken from a Normal Population,”
*Biometrica*17, pp. 364–387.Google Scholar - Venezia, Itzhak (1985), “On the Statistical Origins of the Learning Curve,”
*European Journal of Operational Research**19*, pp. 191–200.CrossRefGoogle Scholar - Weibull, E. H. Waloddi (1939), “A Statistical Theory of the Strength of Materials,”
*Ingeniors Vetenskaps Akademiens Nandlíngar Nr*.*151*.Google Scholar - Womer, N. Keith (1979), “Learning Curves, Production Rate, and Program Costs,”
*Management Science**25*, pp. 312–319.CrossRefGoogle Scholar - Womer, N. Keith (1984), “Estimating Learning Curves from Aggregate Monthly Data,”
*Management Science**30*, pp. 982–992.CrossRefGoogle Scholar - Womer, N. Keith (1984), & Katsuaki Terasawa (forthcoming), “The Effect of Defense Program Uncertainty on Cost, Schedule, and Capital Investment,”
*Journal of Productivity Analysis.*Google Scholar

## Copyright information

© Springer-Verlag New York, Inc. 1989