Question

Explain and define. Explain and define the difference between a convex set, a convex function, and a convex preference. How should the utility function be if the preferences are convex?

Answer 1

1> Convex set is a concept that says if a and b belongs to the set S, then all elements between a and b belongs to S as well. In mathematical terms, if

2> A convex function is such a function where the line segment joining any two points lie above or on the graph.

In mathematical terms, it is

3> a convex preference is a type of preference where an individual prefers average over extremes.

In mathematical terms,

where and , then we have for every :

.

4> Utility function will be quasi-concave if the preference is convex. This means for any convex set S,

and