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Problem Set 2

ECON6001/6701 Microeconomic Analysis 1

Exercise 1. Let X = {x, y, z} and B = {{x, y}} and C({x; y}) = {x}. Show that if (B; C) satisfies WARP, then y / C({x, y, z}).

Exercise 2. A selection of problems to get familiar with indifference curves/sets. Assume X =R+2 .

1) Draw some indifference curves of a DM who likes more of G1 but less of G2.

2) As above, but this time ensure that convex preferences are represented. (For convex prefer-ences, for any pair of indifferent bundles, the average bundle is strictly preferred to either in the original pair.)

3) In the various panels below, I have drawn some indifference curves - a set in case of (C), the blue band. For each of these, explain which of the assumptions: Completeness, Transitivity, Continuity, Local Non-satiation, Convexity and Monotonicity are violated.

4) In comparing a pair of bundles of fruit (apples and oranges), the consumer strictly prefers the bundle that has more fruit in total. If both bundles have the same number of fruit, she strictly prefers the one with more oranges. Draw some indifference curves for this consumer. What property does this preference relation violate?

Exercise 3. The following describes some choice scenarios for a DM (Decision Maker) where you are asked to think about compactness etc. of choice sets

1. A DM can work up to 24 hours in a day, except that she needs must exit the building at midnight. She can also consume "food" during the day.

a) What is the consumption set? Is it closed? Is it bounded?

b) If we specify a wage rate w >0 and a price for food p>0, what might a typical choice set look like? Will it be compact?

2. (Don't worry if you are unable to do this problem. Give it a try!)

There are three possible outcomes : "nice", "not bad" and "terrible". An alternative for the DM is a lottery that results in exactly one of these three outcomes with give probabilities. What is the consumption set? Is it closed? Is it bounded?

Exercise 4. Consider Theorem 1, Page 7, Lec2 Slides. We will prove this in Lec 3. For now provide counter-examples to this result when the assumption of B being convex or the preferences being strictly convex are dropped.

Note. If you are in M.EcA. (M.Ec Analysis) degree or planning to swich to that at some point, I urge you to do Problems 1 - 1.16, end of Chapter 1 problems from Jehle & Reny.