I got a utility function with convex Indifferences curves and therefore convex preferences. Convexity of preferences implies Quasi-concavity. I would like to know if there is a relation between convexity of the preferences and Local non satiation. Does Concavity or quasi-concavity imply local non satiation?
Thank you
Even strict concavity of utility function does not imply local non-satiation. For example: $u(x, y) = -x^2 - y^2$. It is strictly concave because $x^2 + y^2$ is strictly convex. It does not satisfy local non-satiation because $(0,0)$ is the bliss point.
No. Any constant utility function is concave and, hence, quasi-concave and is representing preferences that are as far from being locally non-satiated as can be.
However, strict quasi-concavity (and, therefore, also strict concavity) together with continuity implies that preferences can have at most one satiation-point and that local non-satiation (whichis defined in terms of a for-all-statement) only fails there. To see this, note first that if there were two global satiation points, then any proper convex combination would be even better, which is impossible. To see that local non-satiation can't fail anywhere else, note that any movement towards a better bundle must be an improvement by strong convexity of preferences (which followsfrom strict convexity and continuity.)
It is worth pointing out that the first welfare theorem holds when preferences are locally non-satiated with the possible exception of one satiation point.
