An explicit formula of Hessian determinants of composite functions and its applications

Bang-Yen Chen

The determinants of Hessian matrices of differentiable functions play important roles in many areas in mathematics. In practice it can be difficult to compute the Hessian determinants for functions with many variables. In this article we derive a very simple explicit formula for the Hessian determinants of composite functions of the form: $$f({\bf x})=F(h_{1}(x_{1})+\cdots+ h_{n}(x_{n})).$$ Several applications of the Hessian determinant formula to production functions in microeconomics are also given in this article.