Log-concave probability and its applications

  • Mark Bagnoli
  • Ted Bergstrom
Conference paper
Part of the Studies in Economic Theory book series (ECON.THEORY, volume 26)


In many applications, assumptions about the log-concavity of a probability distribution allow just enough special structure to yield a workable theory. This paper catalogs a series of theorems relating log-concavity and/or log-convexity of probability density functions, distribution functions, reliability functions, and their integrals. We list a large number of commonly-used probability distributions and report the log-concavity or log-convexity of their density functions and their integrals. We also discuss a variety of applications of log-concavity that have appeared in the literature.

Keywords and Phrases

Log-concavity Reliability Hazard functions Probability distributions Failure rates Costly appraisals Mean residual lifetime 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akerlof, G.: The market for ‘lemons’: quality uncertainty and the market mechanism. Quarterly Journal of Economics 84, 599–617 (1970)Google Scholar
  2. 2.
    An, M.Y.: Log-concave probability distributions: theory and statistical testing. Technical Report, Economics Department, Duke University, Durham, NC 27708–0097 (1995)Google Scholar
  3. 3.
    An, M.Y.: Logconcavity versus logconvexity: a complete characterization. Journal of Economic Theory 80(2), 350–369 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Anderson, S., de Palma, A., Thisse, J.-F.: Discrete choice theory and product differentiation. Cambridge, MA: MIT Press 1992Google Scholar
  5. 5.
    Armstrong, M.: Price discrimination by a many-product firm. Review of Economic Studies 66(1), 151–168 (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bagnoli, M., Khanna, N.: Buyers’ and sellers’ agents in the housing market. Journal of Real Estate Finance and Economics 4(2), 147–156 (1991)CrossRefGoogle Scholar
  7. 7.
    Barlow, R., Proschan, F.: Statistical theory of reliability and life testing. New York: Holt Rinehart and Winston 1981Google Scholar
  8. 8.
    Baron, D., Myerson, R.: Regulating a monopolist with unknown costs. Econometrica 50(4), 911–930 (1982)MathSciNetGoogle Scholar
  9. 9.
    Bergstrom, T., Bagnoli, M.: Courtship as a waiting game. Journal of Political Economy 101, 185–202 (1993)CrossRefGoogle Scholar
  10. 10.
    Cameron, C., Segal, J., Songer, D.: Strategic auditing in a political hierarchy: An informational model of the supreme court’s certiori decisions. American Political Science Review 94, 101–116 (2000)Google Scholar
  11. 11.
    Caplin, A., Nalebuff, B.: Aggregation and imperfect competition: on the existence of equilibrium. Econometrica 59(1), 25–60 (1991)MathSciNetGoogle Scholar
  12. 12.
    Caplin, A., Nalebuff, B.: Aggregation and social choice: a mean voter theorem. Econometrica 59(1), 1–24 (1991)MathSciNetGoogle Scholar
  13. 13.
    Chowdry, B., Jegadeesh, N.: Pre-tender offer share acquisition strategy in takeovers. Journal of Financial and Quantitative Analysis 29(1), 117–129 (1994)Google Scholar
  14. 14.
    Corbett, C., de Groote, X.: A supplier’s optimal quantity discount under asymmetric information. Management Science 46(3), 444–450 (2000)CrossRefGoogle Scholar
  15. 15.
    Dierker, E.: Competition for consumers. In: Barnett, W.A., Cornet, B., d’Aspremont, C., Mas-Colell, A. (eds.) Equilibrium theory and applications, pp. 383–402. Cambridge: Cambridge University Press 1991Google Scholar
  16. 16.
    Evans, M., Hastings, N., Peacock, B.: Statistical distributions. New York: Wiley 1993Google Scholar
  17. 17.
    Fang, H., Norman, P.: To bundle or not to bundle. Technical Report 2003-18, University of Wisconsin, Madison, WI (2003)Google Scholar
  18. 18.
    Flinn, C., Heckman, J.: Are unemployment and out of the labor force behaviorally distinct labor force states? Journal of Labor Economics 1, 28–43 (1983)CrossRefGoogle Scholar
  19. 19.
    Goldberger, A.: Abnormal selection bias. In: Karlin, S., Amemiya, T., Goodman, L. (eds.) Studies in econometrics, time series, and multivariate statistics, pp. 67–84. New York: Academic Press 1983Google Scholar
  20. 20.
    Haimanko, O., Le Breton, M., Weber, S.: Transfers in a polarized country: bridging the gap between efficiency and stability. Technical report, Southern Methodist University, Dallas, TX (2003)Google Scholar
  21. 21.
    Heckman, J., Honore, B.: The empirical content of the roy model. Econometrica 58(5), 1121–1149 (1990)Google Scholar
  22. 22.
    Johnson, N., Kotz, S.: Continuous univariate distributions I — distributions in statistics. New York: Wiley 1970Google Scholar
  23. 23.
    Laffont, J.-J., Tirole, J.: Dynamics of incentive contracts. Econometrica 56(5), 1153–1175 (1988)MathSciNetGoogle Scholar
  24. 24.
    Lewis, T., Sappington, D.: Regulating a monopolist with unknown demand. American Economic Review 78(5), 986–998 (1988)Google Scholar
  25. 25.
    Maskin, E. Riley, J.: Monopoly with incomplete information. Rand Journal of Economics 15(2), 282–316 (1984)MathSciNetGoogle Scholar
  26. 26.
    Matthews, S.: Comparing auctions for risk-averse buyers: a buyer’s point of view. Econometrica 55(3), 633–646 (1987)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Muth, E.: Reliability models with positive memory derived from the mean residual life function. In: Tsokos, C., Shimi, I. (eds.) Theory and applications of reliability, vol. II, pp. 401–436. New York: Academic Press 1977Google Scholar
  28. 28.
    Myerson, R., Satterthwaite, M.: Efficient mechanisms for bilateral trading. Journal of Economic Theory 28, 265–281 (1983)MathSciNetGoogle Scholar
  29. 29.
    Noldeke, G., Samuelson, L.: Strategic choice handicaps when females value net viability. Journal of Theoretical Biology 221(1), 53–59 (2003)PubMedGoogle Scholar
  30. 30.
    Patel, J.K., Kapadia, C.H., Owen, D.B.: Handbook of statistical distributions. New York: Marcel Dekker 1976Google Scholar
  31. 31.
    Patil, G.P., Boswell, M.T., Ratnaparkhi, M.V.: Dictionary and classified bibliography of statistical distributions in scientific work vol. 2, continuous univariate models. Fairland, MD: International Cooperative Publishing House 1984Google Scholar
  32. 32.
    Prèkopa, A.: On logarithmic concave measures and functions. Act. Sci. Math. (Szeged) 34, 335–343 (1973)zbMATHGoogle Scholar
  33. 33.
    Proschan, F.: Peakedness of convex combinations. Annals of Mathematical Statistics 36(6), 1703–1706 (1965)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Riordan, M., Sappington, D.: Second sourcing. Rand Journal of Economics 20(1), 41–58 (1989)Google Scholar
  35. 35.
    Rob, R.: Pollution claims under private information. Journal of Economic Theory 47, 307–333 (1989)CrossRefzbMATHGoogle Scholar
  36. 36.
    Segal, I.: Optimal pricing mechanisms with unknown demand. American Economic Review 93(3), 509–529 (2003)Google Scholar
  37. 37.
    Swarz, G.: The mean residual lifetime function. IEEE Transactions in Reliability 26, 108–109 (1973)Google Scholar
  38. 38.
    Verrecchia, R.: Discretionary disclosure and information quality. Journal of Accounting and Economics 12(4), 179–194 (1990)CrossRefGoogle Scholar
  39. 39.
    Verrecchia, R.: Essays on disclosure. Journal of Accounting and Economics 32(1–3), 97–180 (2000)Google Scholar
  40. 40.
    Weber, S.: On hierarchical spatial competition. Reviewof Economic Studies 59(2), 407–425 (1992)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Mark Bagnoli
    • 1
  • Ted Bergstrom
    • 2
  1. 1.Department of AccountingPurdue UniversityWest LafayetteUSA
  2. 2.Department of EconomicsUC Santa BarbaraSanta BarbaraUSA

Personalised recommendations