The Degree of Monopoly and Multivariable Sales Policies

  • Klaus Spremann


The degree of monopoly for multivariable sales policies and multipart tariffs, where these output decisions may be corner solutions of the admissible set, and where there may be kinks in the demand or cost functions is analyzed within a general model. A monopoly is understood as a situation where an agent (the firm) affects the aggregate welfare of the economy by his output decisions via external effects. These external effects and the marginal loss of welfare due to the selfishness of the agent can be measured by two real numbers ω+, ω-, which have the meaning of indices of aggression against and consideration of the society’s objective. Hence the degree of monopoly is defined as a function, which assigns to every situation a pair (ω+, ω-). The determination of this pair (ω+, ω-) for a given situation requires a special linearization technique for sets and functional s which is termed homogeneous approximation of the situation under consideration. The main result states the failure of efficiency of situations, which reveal (ω+ > ω-); and examples illustrate that almost all situations with multivariable sales policies are non-efficient in the following sense : There is a slight modification of the agent’s policy which is admissible and increases both his own utility (profit) and the society’s welfare. Conditions for such a “non-tangency” of iso-profit and iso-welfare loci are derived in real Banach spaces and expressed in terms of (the value of) the degree of monopoly (ω+, ω-).


Price Discrimination Real Banach Space Aggregate Welfare Output Decision Homogeneous Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.S. BAIN: The Profit Rate as a Measure of Monopoly Power, Quarterly Journal of Economics, Vol. LV (1941), 271–293.CrossRefGoogle Scholar
  2. 2.
    J.S. BAIN: Measurements of the Degree of Monopoly : a Note, Economica, Vol. 10(1943), 66.CrossRefGoogle Scholar
  3. 3.
    W.J. BAUMOL and D. BRADFORD: Optimal Departures From Marginal Cost Pricing, American Economic Review, Vol. 60 (1970), 265–283.Google Scholar
  4. 4.
    J.M. BUCHANAN and W.C. STUBBLEBINE: Externality, Economica, Vol. 29(1962), 371–384.CrossRefGoogle Scholar
  5. 5.
    J. DIEUDONNE: Foundations of Modern Analysis, Vol. 10–1 in Pure and Applied Mathematics, Academic Press, New York 1969, ch. VIII.Google Scholar
  6. 6.
    I.V. GIRSANOV: Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, Vol. 67, Springer-Verlag, Berlin 1972, p.50 f.CrossRefGoogle Scholar
  7. 7.
    O.C. HERFINDAHL: Concentration in the Steel Industry, doctoral dissertation, Columbia University, New York, 1950.Google Scholar
  8. 8.
    M.R. HESTENES : Calculus of Variations and Optimal Control Theory, Applied Mathematics Series, Wiley, New York, 1966.Google Scholar
  9. 9.
    Y.-K. NG and M. WEISSER: Optimal Pricing with a Budget Constraint The Case of the Two-part Tariff, Review of Economic Studies, Vol. 41 (July 1974) 137, 337–345.CrossRefGoogle Scholar
  10. 10.
    J.R. HICKS: The rehabilitation of Consumers’ Surplus, The Review of Economic Studies, Vol. 9(1941), 108–116.CrossRefGoogle Scholar
  11. 11.
    R.W. KILPATRICK : The Choice Among Alternative Measures of Industrial Concentration, Review of Economics and Statistics, (May 1967), 258–260.Google Scholar
  12. 12.
    P.J. LAURENT: Approximation et optimisation. Collection Einseignement des sciences, Vol. 13, Hermann, Paris, 1972.Google Scholar
  13. 13.
    A.P. LERNER: The Concept of Monopoly and the Measurement of Monopoly Power, The Review of Economic Studies, Vol. 1(1934), 157–175.CrossRefGoogle Scholar
  14. 14.
    S.C. LITTLECHILD : Two-part Tariffs and Consumption Externalities, Bell Journal of Economics and Management Science, Autumn 1975, 661–670.Google Scholar
  15. 15.
    L.A. LJUSTERNIK and W.I. SOBOLEW : Elemente der Funktionalanalysis, Mathematische Lehrbücher Band VIII, Akademie-Verlag, Berlin, 1968.Google Scholar
  16. 16.
    F. MACHLUP: Monopoly and Competition : A Classification of Market Positions, American Economic Review, Vol. XXVI1(1937), 445–451.Google Scholar
  17. 17.
    R.A. MEYER : Monopoly Pricing Structures with Imperfect Discrimination, Bell Journal of Economics and Management Science, Autumn 1976,Google Scholar
  18. 18.
    T. MORGAN : A Measure of Monopoly in Selling, Quarterly Journal of Economics, Vol. LX (1946), 461–463.CrossRefGoogle Scholar
  19. 19.
    M.Z. NASHED: Differentiability and Related Properties of Nonlinear Operators : Some Aspects of the Role of Differentials in Nonlinear Functional Analysis, in : L.B. BELL (ed.) : Nonlinear Functional Analysis, Publication No. 26 of the Mathematics Research Center of the University of Wisconsin, Academic Press, New York, 1971, 103–309.Google Scholar
  20. 20.
    W. OI: A Disneyland DiVemma : Tωo Part Tariffs for a Mickey Mouse Monopoly, Quarterly Journal of Ecqnomics, Vol. 85 (1971), 77–96.CrossRefGoogle Scholar
  21. 21.
    A.G. PAPANDREOU : Market Structure and Monopoly Power, American Economic Review, Vol. XXXIX(1949), 883–897.Google Scholar
  22. 22.
    T. RADER: Equivalence of Consumer Surplus, the Divisia Index of Output, and Eisenberg’s Addilog Social Utility, Journal of Economic Theory, Vol. 13(1976), 58–66.CrossRefGoogle Scholar
  23. 23.
    F. RAMSEY: A Contribution to the Theory of Taxation, Economic Journal, Vol. 37 (March 1927), 47–61.CrossRefGoogle Scholar
  24. 24.
    K.W. ROTHSCHILD: The Degree of Monopoly, Economica, Vol. 9(1942), 21–39.CrossRefGoogle Scholar
  25. 25.
    F.M. SCHERER: Industrial Market Structure and Economic Performance, Rand Mc Nally, Chicago, 1970, 50–71.Google Scholar
  26. 26.
    K. SPREMANN: Über Vektormaximierung und Analyse der Gewichtung von Subzielen, in : W. OETTLI and K. RITTER (eds.) : Optimization and Operations Research, Lecture Notes in Economics and Mathematical Systems, Vol. 117, Springer-Verlag, Berlin, 1976, 283–296.CrossRefGoogle Scholar
  27. 27.
    K. SPREMANN: On Welfare Implications and Efficiency of First Fee Pricing, Discussion Paper des Instituts für Wirtschaftstheorie und Operations Research der Universität Karlsruhe, No. 78, (Feb. 1977).Google Scholar
  28. 28.
    R.A. TAPIA: The Differentiation and Integration of Nonlinear Operators, in : L.B. BELL (ed.) : Nonlinear Functional Analysis, Publication No. 26 of the Mathematics Research Center of the University of Wisconsin, Academic Press, New York, 1971, 45–102.Google Scholar
  29. 29.
    R. TRIFFIN: Monopolistic Competition and General Equilibrium Theory, Cambridge, 1940.Google Scholar
  30. 30.
    P.P. VARAIYA: Nonlinear Programming in Banach Space, SI AM Journal on Applied Mathematics, Vol. 15 (1967) 2, 284–293.CrossRefGoogle Scholar
  31. 31.
    R.D. WILLIG: Consumer’s Surplus Without Apology, American Economic Review, Vol. 66 (Sept. 1976), 589–597.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  • Klaus Spremann

There are no affiliations available

Personalised recommendations