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

TL;DR

This paper classifies homothetic functions satisfying the homogeneous Monge-Ampère equation and explores their applications in economic production models, providing a mathematical foundation for understanding certain economic functions.

Contribution

It offers a complete classification of homothetic solutions to the homogeneous Monge-Ampère equation with applications to economic production functions.

Findings

Classification of homothetic solutions to the Monge-Ampère equation

Applications to economic production models

Insights into marginal rate of substitution in economics

Abstract

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.