Solutions to homogeneous Monge-Amp\`ere equations of homothetic functions and their applications to production models in economics

Mathematically, a homothetic function is a function of the form $f({\bf x})=F(h(x_1,...,x_n))$, where $h$ is a homogeneous function of any degree $d\ne 0$ and $F$ is a monotonically increasing function. In economics homothetic functions are production functions whose marginal technical rate of substitution is homogeneous of degree zero. In this paper we classify homothetic functions satisfying the homogeneous Monge-Amp\`ere equation. Several applications to production models in economics will also be given.