MATH2003J, OPTIMIZATION IN ECONOMICS,

BDIC 2023/2024, SPRING

Problem Sheet 12

Question 1:

Determine whether each of the following is true or false. Justify your answers.

(1) If S1 and S2 are two convex sets in R2, then their union

S1 ∪ S2 = {x ∈ R 2 ∶ x ∈ S1 or x ∈ S2}

must be a convex set in R2.

(2) If f and g are two concave functions on R n , then the function f + g is concave on R n .

Question 2:

Sketch the following sets and decide which are convex. Justify your answers.

(a) {(x, y) ∈ R2 ∣ x, y ≥ 0 and x + y < 1}

(b) {(x, y) ∈ R2 ∣ 1 ≤ x2 + y2 ≤ 4}

(c) {x ∈ R ∣ ∣x∣ > 1}

Question 3:

Consider the function C(x, y) = 100/1x2 − 10x + 300/1y3 − 9y for x, y ≥ 0.

(a) Examine if C is concave, convex or neither of them.

(b) Find the critical points of C and show that at these points C attains a minimum.