Is it possible?
For example, is it possible for a function $f(x,y)$
- to satisfy the sufficient condition for quasi concavity -- (the sufficient condition on the determinants of the leading principle minors of the bordered hessian) -- for $(x,y)\in (4,5)\times (2,3)$,
- and to also satisfy them for $(x,y) \in (4,5)\times (0,2)$,
- but the function to not be quasi concave over $(4,5)\times (0,3)$?
Some comments:
- The intervals I used in my example don't matter, I just used any numbers in $(0,\infty)$
- Graphically, I think I can imagine such a function. For example, suppose the function is increasing in $y$ for $y\in (2,3)$ but decreasing in $y$ for $y\in (0,2)$.
- In such a case I think the "upper contour sets" will be in different directions in the two regions (by different directions I mean in one region higher values of $y$ are better and in the other lower values of $y$ are better)
If the function's bordered hessian satisfies the sufficient condition for quasi-concavity over two adjacent regions, but the function is not quasi-concave over the union of the regions, what does this mean for the derivative of the function?
- To elaborate, we can write the sufficient conditions for quasi-concavity in terms of derivatives. In the situation I have described I think the condition must be violated at the border between the two regions. Doesn't this mean that the first or second derivatives must have some sort of discontinuity at the border of the two regions?
lastly, if one of the regions was a closed interval -- i.e. $[0,2]$, then would satisfying the sufficient condition for quasi-concavity over the two adjacent regions imply quasi-concavity over the union of the regions?