2.10 The Cobb-Douglas function One of the most important functions we will encounter in this book is the Cobb-Douglas function: y = (x)"(x), where a and B 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 = c (where is any positive constant) is convex and there- fore that the set of points for which y > c is a convex set. c. Show that if a + B > 1, then the Cobb-Douglas func- tion 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. 2.10 The Cobb-Douglas function One of the most important functions we will encounter in this book is the Cobb-Douglas function: y = (x)"(x), where a and B 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 = c (where is any positive constant) is convex and there- fore that the set of points for which y > c is a convex set. c. Show that if a + B > 1, then the Cobb-Douglas func- tion 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