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Eco 600E Advanced Microeconomics I Term: Spring (1st), 2009

Lecturer: Yosuke Yasuda

Problem Set 1 Due in class on May 7

1. Question 1 (10 points)

Prove the followings (DeMorgan’s Law):

(S \ T )^c = S^{c [Tc} (S [ T )^c = S^{c \}T^c

Hint: You should use the de…nitions of union, intersection, and complement of sets. Drawing …gures (Venn diagrams) is not enough.

2. Question 2 (10 points)

Let A and B be convex sets in R². Then, answer the following questions. (a) Construct an example such that A [ B is not a convex set.

(b) Show that A \ B must be a convex set. 3. Question 3 (10 points)

Let u : R^{K !} R be a concave function. Then, show that u is also a quasi-concave function.

4. Question 4 (20 points)

Suppose % is a preference relation on X. Then, show the followings. (a) Re‡exive: For any x 2 X, x x.

(b) Transitive 1: For any x; y; z 2 X, if x y and y z, then x z. (c) Transitive 2: For any x; y; z 2 X, if x y and y z, then x z. (d) Transitive 3:For any x; y; z 2 X, if x y and y % z, then x % z. where and are de…ned as follows:

a b , a% b and b % a a b , a% b and not b % a

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5. Question 5 (20 points)

Let X = R²+. Assume that a preference relation % satis…es the following three prop- erties:

Additive: (a1; a2) % (b1; b2) implies that (a1+ t; a2+ s) % (b1+ t; b2+ s) for all t and s.

Strictly monotone: If a1 b1 and a2 b2, then ; in addition, if either a1 > b1 or a2 > b2, then (a1; a2) (b1; b2).

Continuous

(a) Show that if % is represented by a linear utility function, i.e., u(x1; x2) = x1+ x2

with ; >0, then % satis…es the above three properties.

(b) Find the preference relation that is 1) Additive and Strictly monotone but not Continuous, and 2) Strictly monotone and Continuous but not Additive.

6. Question 6 (10 points)

We say that a preference relation % is homothetic if x % y implies x % y for all 0.

Show that if a consumer has a homothetic preference relation, then her demand function is homogeneous of degree one in !. That is, x(p; !) = x(p; !) for any ^>0. 7. Question 7 (20 points)

Suppose there are three types of goods, and the consumer chooses di¤erent bundles for three di¤erent price vectors in the following way. Namely, she chooses bundle xⁱ at prices pⁱ, i = 1; 2; 3, where

p¹ = 0

@ 1 1 2

1

A; p² ₌ 0

@ 1 1 1

1

A; p³ ₌ 0

@ 1 2 1

1 A;

x¹ = 0

@ 5 19

9 1

A; x² ₌ 0

@ 12 12 12

1

A; x³ ₌ 0

@ 27 11 1

1 A:

In each case, the amount of expenditure is equal to her initial wealth, i.e., !ⁱ = pⁱxⁱ for i = 1; 2; 3.

(a) Show that the above data satisfy the Weak Axiom of revealed preference. (b) Show that this consumer’s behavior cannot be fully rationalized.

Hint: Assume there is some preference relation % that fully rationalizes the above data, and verify that % fails to satisfy transitivity.

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