Homothetic functions revisited |
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Authors: | PO Lindberg E Anders Eriksson Lars-Göran Mattsson |
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Institution: | Department of Mathematics, Link?ping University, 581 83 Link?ping, SWEDEN, SE Division of Defence Analysis, National Defence Research Establishment,172 90 Stockholm, SWEDEN, SE Department of Infrastructure and Planning, Royal Institute of Technology, 100 44 Stockholm, SWEDEN (e-mail: LGM@infra.kth.se), SE
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Abstract: | Summary. 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.
Received: November 5, 1999; revised version: October 31, 2000 |
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Keywords: | and Phrases: Homotheticity Homothetic functions Homogeneous functions Functional dependence |
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