THE UNIVERSITY OF HONG KONG Faculty of Business and Economics ECON6011 Microeconomic Theory Problem Set 4 Due: October 6, 2 pm. 1. Let {pk} ⊆ P converges to p ∈ P . Let Y ⊆ Rn be standard and y(·) be the corre- sponding supply function. Show that there exists a compact set C such that y(pk) ∈ C for all k. 2. Consider R2, i.e., there are only two goods. Let Y 1 = {(x, y) ∈ R2|y ≤ −x and x, y ≤ 1} and Y 2 = {(x, y) ∈ R2|y ≤ 1 + 1 x−1 and x < 1}. (i) Show that Y 2 is standard; (ii) Show that Y 1 is nonempty, closed, convex, and satisfies free disposal, possibility of inaction, and boundedness from above; (iii) Show that Y 1 is not strictly convex; (iv) Show that Y 1 + Y 2 is not standard. 3. Let Y be standard and x ∈ Rn. If there exists p ∈ Rn, c ∈ R such that p · x > c > p · y for all y ∈ Y . Then there exists p∗ ∈ P and c∗ ≥ 0 such that p∗ ·x > c∗ > p∗ · y for all y ∈ Y . (Hint: use y). 1