Second-order condition

Introduction 1: The function $z(x,y)$ is a function of two variables. Hence, we cannot speak of a the derivative of $z(x,y)$, but we have to specify whether we mean the partial derivative with respect to $x$ or to $y$, hence $z'_x(x,y)$ or $z'_y(x,y)$.

Introduction 2: A function of one variable is convex if the derivative increases, hence if the second-order derivative is non-negative. A similar reasoning holds for functions of two variables. We have to consider, however, the second-order partial derivatives of $z(x,y)$, hence $z''_{xx}(x,y)$, $z''_{yy}(x,y)$ and $z''_{xy}(x,y)=z''_{yx}(x,y)$.

Introduction 3: Whether a function of two variables is convex or concave depends on the sign of the criterion function
$$C(x,y)=z''_{xx}(x,y)z''_{yy}(x,y)-(z''_{xy}(x,y))^2.$$

Theorem:

• If $C(x,y)\geq 0$, $z_{xx}''(x,y)\geq 0$ and $z_{yy}''(x,y)\geq 0$ on a part of the domain, then the function $z(x,y)$ is convex on that part of the domain.
• If $C(x,y)\geq 0$, $z_{xx}''(x,y)\leq 0$ and $z_{yy}''(x,y)\leq 0$ on a part of the domain, then the function $z(x,y)$ is concave on that part of the domain.

A function such that $C(x,y)< 0$ on part of the domain, is neither convex nor concave on that part of the domain.