On the Curvature of Homogeneous Functions

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.