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### Practice Questions for Midterm

Subject: Advanced Microeconomics I (ECO600E) Professor: Yosuke YASUDA

1. True or False

Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.

(a) A binary relation % is said to be a preference relation if it is “complete” and

“transitive.”

(b) If consumer’s choice satis…es the weak axiom of revealed preferences, we can always construct a utility function which is consistent with such choice behav- iour.

(c) If a consumer problem has a solution, then it must be unique whenever the consumer’s preference relation is convex.

(d) Suppose % is represented by utility function u( ). Then, u( ) is concave if and only if % is convex.

2. Sets

Prove the followings (DeMorgan’s Law):

(S \ T )c = Sc[ Tc (S [ T )c = Sc\ Tc

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

3. Cocavity

Construct a monotone function f : R2+ ! R, which is quasi-concave but NOT a concave function.

4. Preferences

Suppose % is a preference relation on X. That is, % satis…es completeness and transitivity. Then, show the followings.

(a) For any x; y; z 2 X, if x y and y % z, then x % z. (b) For any x; y; z 2 X, if x y and y z, then x z.

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where and are de…ned as follows:

a b , a % b and b % a a b , a % b and not b % a 5. Choice

Consider a consumer problem. Suppose that a choice function x(p; !) satis…es Walras’s law and WA. Then, show that x(p; !) is homogeneous of degree zero. 6. Lagrange’s Method

You have two …nal exams upcoming, Mathematics (M) and Japanese (J), and have to decide how to allocate your time to study each subject. After eating, sleeping, exercising, and maintaining some human contact, you will have T hours each day in which to study for your exams. You have …gured out that your grade point average (G) from your two courses takes the form

G= 4 7(2

pJ +pM),

where J (/ M ) is the number of hours per day spent studying for Japanese (/ Math- ematics). You only care about your GPA. Then, answer the following questions.

(a) What is your optimal allocation of study time?

(b) Suppose T = 10. If you follow this optimal strategy, what will be your GPA? 7. Kuhn-Tucker Condition

Consider the following problem: Maximize W (x; y) = ln(x) + ln(y) subject to the following constraints:

x a; y 0; x + y 10

where a is a non-negative parameter. Then, answer the following questions.

(a) Solve this problem by using Kuhn-Tucker conditions (you can assume second order conditions are satis…ed), and derive the maximum derive function M (a). (b) Now substitute a = 2. Derive the bordered Hessian and verify that your

solution is a global maximum.

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