Abstract
Let f(x) be a continuous, homothetic function defined in a connected cone D. Assume that f is strictly increasing along each ray in D, i.e. for each x0 ≠ 0 in D, f(tx0) is a strictly increasing function of t. Then there exist a homogeneous function g and a strictly increasing function F such that f(x) = F(g(x)) for all x in D
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References
Most of the formulas are standard and can be found in almost any calculus text, e.g. Edwards and Penney (1998), or Sydsæter and Hammond (2005). For supergradients and differentiability, see e.g. Sydsæter et al. (2005). For properties of homothetic functions, see Simon and Blume (1994), Shephard (1970), and Førsund (1975).
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Sydsæter, K., Strøm, A., Berck, P. (2010). Partial derivatives. In: Economists’ Mathematical Manual. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28518-2_4
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DOI: https://doi.org/10.1007/978-3-540-28518-2_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26088-2
Online ISBN: 978-3-540-28518-2
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