2.9 One of the most important functions we will encounter in this book is the Cobb-Douglas function:

y = (x where a and P are positive constants that are each less than one.

a. Show that this function is quasi-concave using a "brute force" method by applying Equa tion 2.107.

b. Show that the Cobb-Douglas function is quasi-concave by showing that the any contour line of the form y = c (where c is any positive constant) is convex and therefore that the set of points for which y > c is a convex set.

c. Show that if a + y8 > 1 then the Cobb-Douglas function is not concave (thereby illus trating that not all quasi-concave functions are concave). (Note: The Cobb-Douglas func tion is discussed further in the Extensions to this chapter.)