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### Problem Set 1: Posted on October 14

Advanced Microeconomics I (Fall, 1st, 2014) 1. Question 1 (6 points)

Prove each of the following statements:

(a) The intersection of any pair of open sets is an open set.

(b) The union of any (possibly infinite) collection of open sets is open.

(c) The intersection of any (possibly infinite) collection of closed sets is closed. (You can use (b) and De Morgan’s Law without proofs.)

2. Question 2 (6 points)

Let A and B be convex sets in R2. Then, answer the following questions. (a) Show that A ∩ B must be a convex set.

(b) Construct an example such that A ∪ B is not a convex set. (c) Show that A − B defined below is a convex set.

A− B := {x | x = a − b, a ∈ A, b ∈ B}.

3. Question 3 (3 points)

Let A and B be two sets in domain D, and suppose that B ⊂ A. Prove that f(B) ⊂ f(A) for any mapping f : D → R.

4. Question 4 (6 points)

A utility function U (x) : X → R is called quasi-linear if it can be written as U(x) = αy + V (z),

with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V(·) is a real valued function. The consumption set X = Rn+1+ .

(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave.

5. Question 5 (4 points)

A consumer has the following concave utility function, U(x) = x1x2+x3x4.

Solve for the consumer’s choice as a function of her income ω and prices p1, . . . , p4.

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