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

y 5 1x12α 1x22 β

, where α and β are positive constants that are each less than 1.

a. Show that this function is quasi-concave using a “brute force” method by applying Equation 2.100

.

b. Show that the Cobb–Douglas function is quasi-concave by showing that any contour line of the form y 5 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 α 1 β . 1, then the Cobb–Douglas function is not concave (thereby illustrating again that not all quasi-concave functions are concave).

Note: The Cobb–Douglas function is discussed further in the Extensions to this chapter.