Homothetic Functions Revisited

• Published in: Economic theory Vol. 19; no. 2; pp. 417 - 427
• Main Authors: Lindberg, P. O., Eriksson, E. Anders, Mattsson, Lars-Göran
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer-Verlag 01.02.2002; Springer Nature B.V
• Subjects: Continuous functions; Differentiable functions; Economic methodology; Economic theory; Economics; Exposita Notes; functional dependence; homogeneous functions; homothetic functions; homotheticity; Marginal rate of substitution; Mathematical constants; Mathematical functions; Mathematical monotonicity; Methodology; Production functions; TECHNOLOGY; TEKNIKVETENSKAP
• ISSN: 0938-2259; 1432-0479; 1432-0479
• DOI: 10.1007/PL00004220

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.