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Journal of Optimization Theory and Applications

On the Curvature of Homogeneous Function

Journal Article
Hjertstrand, Per (2023). “On the Curvature of Homogeneous Function”. Journal of Optimization Theory and Applications 198(1), 215–223.

Per Hjertstrand

Consider a quasiconcave, upper semicontinuous and homogeneous of degree $\gamma$ function $f$. This paper shows that the reciprocal of the degree of homogeneity, $1/\gamma$, 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 $\gamma \leq -1$ or $\gamma \geq 0$, concave if $0 \leq \gamma \leq 1$ and logconcave if $ \gamma \geq 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.