Problem Set 1: Due on April 26
Advanced Microeconomics I (Spring, 1st, 2012)
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 (2 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 (2 points)
Let u : R2 → R be a concave function. Then, show that u is also quasi-concave. 5. Question 5 (4 points)
Suppose that u(x1, x2) and v(x1, x2) are utility functions.
(a) Show that if u(x1, x2) and v(x1, x2) are both homogeneous of degree r, then s(x1, x2) := u(x1, x2) + v(x1, x2) is also homogeneous of degree r.
(b) Show that if u(x1, x2) and v(x1, x2) are quasi-concave, then m(x1, x2) := min{u(x1, x2), v(x1, x2)} is also quasi-concave.
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