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ECON6011 Microeconomic Theory

Problem Set 4

        1. Let {p^k} ⊆ 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(p^k) ∈ 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).