The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Quasi-Concavity

  • J.-P. Crouzeix
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1863

Abstract

A real function f defined on a convex subset C of a linear space E is said to be quasi-concave if
$$ x,y\in C,\kern0.5em t\in \left[0,1\right]\Rightarrow f\left( tx+\left(1-t\right)y\right)\ge \operatorname{Min}\left[f(x),f(y)\right]. $$
A function g is said to be quasi-convex if – g is quasi-concave. Concave functions are quasi-concave, convex functions are quasi-convex.

JEL Classifications

D0 
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Bibliography

  1. An important and up to date discussion of quasiconcavity and related topics with their applications for economics as well as for mathematical programming can be found in Generalized concavity in optimization and economics, a collection of papers by several authors edited by S. Schaible and W.T. Ziemba (New York: Academic Press, 1981).Google Scholar
  2. Arrow, K.J., and A.C. Enthoven. 1961. Quasi-concave programming. Econometrica 29(4): 779–800.CrossRefGoogle Scholar
  3. Crouzeix, J.P. 1983. Duality between direct and indirect utility functions. Journal of Mathematical Economics 12(2): 149–165.CrossRefGoogle Scholar
  4. Crouzeix, J.P., and P.O. Lindberg. 1986. Additively decomposed quasi-convex functions. Mathematical Programming 35(1): 42–57.CrossRefGoogle Scholar
  5. Debreu, G. 1976. Least concave utility functions. Journal of Mathematical Economics 3(2): 121–129.CrossRefGoogle Scholar
  6. Debreu, G., and T.C. Koopmans. 1982. Additively decomposed quasi-convex functions. Mathematical Programming 24(1): 1–38.CrossRefGoogle Scholar
  7. De Finetti, B. 1949. Sulle stratificazioni convesse. Annali di matematica pura ed applicata, Series IV, 30: 173–183.Google Scholar
  8. Diewert, W.E. 1981. Generalized concavity in economics. In Generalized concavity in optimization and economics, ed. S. Schaible and W.T. Ziemba. New York: Academic Press.Google Scholar
  9. Fenchel, W. 1953. Convex cones, sets and functions. Mimeo, Princeton University.Google Scholar
  10. Kannai, Y. 1981. Concave utility functions. In Generalized concavity in optimization and economics, ed. S. Schaible and W.T. Ziemba. New York: Academic Press.Google Scholar
  11. Mangasarian, O.L. 1965. Pseudo-convex functions. SIAM Journal on Control 3(2): 281–290.Google Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • J.-P. Crouzeix
    • 1
  1. 1.