On the Curvature of Homogeneous Function

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.