Consider a consumer whose consumption set is X=R+l. a) Define a quasi-concave utility function u() on the consumption set X=R+l. [5 marks] b) Show that a utility function u() that represents a rational preference relation on the consumption set X=R+l is quasi-concave if and only if the preference relation is convex. [5 marks] c) Show that the Walrasian budget set B(p,w) is convex. [5 marks] c) Suppose that a consumer optimally chooses a consumption bundle from her budget set B(p,w), given her wealth w and the price vector p0. Show that if the consumer's utility function is strictly quasi-concave, then this choice problem has at most one solution. Consider a consumer whose consumption set is X=R+l. a) Define a quasi-concave utility function u() on the consumption set X=R+l. [5 marks] b) Show that a utility function u() that represents a rational preference relation on the consumption set X=R+l is quasi-concave if and only if the preference relation is convex. [5 marks] c) Show that the Walrasian budget set B(p,w) is convex. [5 marks] c) Suppose that a consumer optimally chooses a consumption bundle from her budget set B(p,w), given her wealth w and the price vector p0. Show that if the consumer's utility function is strictly quasi-concave, then this choice problem has at most one solution