On the Curvature of Homogeneous Functions

Per Hjertstrand ()
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Per Hjertstrand: Research Institute of Industrial Economics (IFN)

Journal of Optimization Theory and Applications, 2023, vol. 198, issue 1, No 8, 215-223

Abstract: Abstract Consider a quasiconcave, upper semicontinuous and homogeneous of degree $$\gamma $$ γ function f. This paper shows that the reciprocal of the degree of homogeneity, $$1/\gamma $$ 1 / γ , can be interpreted as a measure of the degree of concavity of f. As a direct implication of this result, it is also shown that f is harmonically concave if $$\gamma \le -1$$ γ ≤ - 1 or $$\gamma \ge 0$$ γ ≥ 0 , concave if $$0\le \gamma \le 1$$ 0 ≤ γ ≤ 1 and logconcave if $$\gamma \ge 0$$ γ ≥ 0 . Some relevant applications to economic theory are given. For example, it is shown that a quasiconcave and homogeneous production function is concave if it displays nonincreasing returns to scale and logconcave if it displays increasing returns to scale.

Keywords: Concavity; Homogeneity; Logconcavity; $$\rho $$ ρ -concavity; Quasiconcavity; Primary 26B25; Secondary 90C99; 91B38 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02249-6

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