Combination of supermodular and submodular functions

Question

Suppose the production function $v(x,y)$ is increasing and submodular in both arguments, and the production function $c(x,y)$ is increasing and supermodular in both arguments ($x,y \geq 0$). Is the difference $d(x,y) = v(x,y)-c(x,y)$ always again submodular? Examples like $c(x,y)=(x+y)^2$ and $v(x,y)=\sqrt{x+y}$ suggest this is true due to $\frac{\partial}{\partial x \partial y} d(x,y) < 0$. Any general proof/contradiction or reference for this? Thanks!

Answer 1

Suppose $f$ is supermodular, i.e. $$f(x\uparrow y)+f(x\downarrow y)\ge f(x)+f(y)$$ Suppose also $g$ is supermodular. Set $h(x)=f(x)+g(x)$, and we have $$h(x\uparrow y)+h(x\downarrow y)=f(x\uparrow y)+g(x\uparrow y)+f(x\downarrow y)+g(x\downarrow y)$$ $$=f(x\uparrow y)+f(x\downarrow y)+g(x\uparrow y)+g(x\downarrow y)$$ $$\ge f(x)+f(y)+g(x)+g(y)=f(x)+g(x)+f(y)+g(y)=h(x)+h(y)$$