In this exercise, we will explore some logical relationships between families of tastes that satisfy different assumptions.

A: Suppose we define a strong and a weak version of convexity as follows: Tastes are said to be strongly convex if, whenever a person with those tastes is indifferent between A and B, she strictly prefers the average of A and B (to A and B). Tastes are said to be weakly convex if, whenever a person with those tastes is indifferent between A and B, the average of A and B is at least as good as A and B for that person.

(a) Let the set of all tastes that satisfy strong convexity be denoted as SC and the set of all tastes that satisfy weak convexity as WC. Which set is contained in the other? (We would, for instance, say that €œWC is contained in SC€ if any taste that satisfies weak convexity also automatically satisfies strong convexity.)

(b) Consider the set of tastes that are contained in one and only one of the two sets defined above.

What must be true about some indifference curves on any indifference map from this newly defined set of tastes?

(c) Suppose you are told the following about 3 people: Person1 strictly prefers bundle A to bundle B whenever A contains more of each and every good than bundle B. If only some goods are represented in greater quantity in A than in B while the remaining goods are represented in equal quantity, then A is at least as good as B for this person. Such tastes are often said to be weakly monotonic. Person 2 likes bundle A strictly better than B whenever at least some goods are represented in greater quantity in A than in B while others may be represented in equal quantity. Such tastes are said to be strongly monotonic. Finally, person 3€™s tastes are such that, for every bundle A, there always exists a bundle B very close to A that is strictly better than A. Such tastes are said to satisfy local nonsatiation. Call the set of tastes that satisfy strict monotonicity SM, the set of tastes that satisfy weak monotoni city WM, and the set of tastes that satisfy local non-satiation L. What is the relationship between these sets? Put differently, is any set contained in any other set?

(d) Give an example of tastes that fall in one and only one of these three sets?

(e) What is true of tastes that are in one and only one of the two sets SM andWM?

B: Below we will consider the logical implications of convexity for utility functions. For the following definitions, 0 ‰¤ Î± ‰¤ 1. A function f R2+ †’R1 is defined to be quasi concave if and only if the following is true: Whenever f

The same type of function is defined to be concave if and only if αf

(a) True or False: All concave functions are quasiconcave but not all quasiconcave functions are concave.

(b) Demonstrate that, if u is a quasiconcave utility function, the tastes represented by u are convex.

(c) Do your conclusions above imply that, if u is a concave utility function, the tastes represented by u are convex?

(d) Demonstrate that, if tastes over two goods are convex, any utility functions that represents those tastesmust be quasiconcave.

(e) Do your conclusions above imply that, if tastes over two goods are convex, any utility function that represents those tastes must be concave?

(f) Do the previous conclusions imply that utility functionswhich are not quasiconcave represent tastes that are not convex?

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