Economic Theory

, Volume 19, Issue 2, pp 417–427 | Cite as

Homothetic functions revisited

  • P.O. Lindberg
  • E. Anders Eriksson
  • Lars-Göran Mattsson
Exposita Notes


This paper examines the proposition that homotheticity is equivalent to the property that (e.g., in the context of a production function) the marginal rate of substitution is constant along any ray from the origin. This claim is made in many places, but hitherto the prerequisites have not been stated explicitly. In the present contribution it is demonstrated that an additional condition is required for the claim to hold. We present a theorem that achieves equivalence by also assuming ‘nowhere ray constancy’. It turns out that this condition is implied by assumptions often made, e.g., in production theory. Further, a complete characterization is given of the class of functions that satisfy ray constant marginal rates of substitution or, somewhat more generally, a condition of ray parallel gradients. In addition to homothetic functions this class contains functions homogeneous of degree 0 (i.e., ray constant) and functions which are homothetic in disjoint regions separated by regions of ray constancy.

Keywords and Phrases: Homotheticity, Homothetic functions, Homogeneous functions, Functional dependence. 
JEL Classification Numbers: C00, D20, D11. 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • P.O. Lindberg
    • 1
  • E. Anders Eriksson
    • 2
  • Lars-Göran Mattsson
    • 3
  1. 1.Department of Mathematics, Linköping University, 581 83 Linköping, SWEDEN SE
  2. 2.Division of Defence Analysis, National Defence Research Establishment,172 90 Stockholm, SWEDEN SE
  3. 3.Department of Infrastructure and Planning, Royal Institute of Technology, 100 44 Stockholm, SWEDEN (e-mail: SE

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