ECON6011 Microeconomic Theory
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ECON6011 Microeconomic Theory
Problem Set 4
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 Y1 = {(x, y) ∈ R2|y ≤ −x and x, y ≤ 1} and Y2 = {(x, y) ∈ R2|y ≤ 1 + and x < 1}.
(i) Show that Y2 is standard;
(ii) Show that Y1 is nonempty, closed, convex, and satisfifies free disposal, possibility of inaction, and boundedness from above;
(iii) Show that Y1 is not strictly convex;
(iv) Show that Y1 + Y2 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).
2021-10-09