MATH2003J, OPTIMIZATION IN ECONOMICS,

BDIC 2023/2024, SPRING

Problem Sheet 13

Question 1:

For which a ∈ R is the function f ∶ R 3 ! R, f(x, y, z) = x 2 + xz + ayz + z 2 convex?

Question 2:

Show that the function f ∶ R 4 ! R, with f(x, y, z, w) = x 2 + y 2 + z 2 + w 2 − 2xw − 4yz is neither convex nor concave on R 4 . Also, give an example of a function g ∶ R 4 ! R which is both convex and concave.

Question 3:

Show that the function f defined for K, L > 0 by

f(K, L) = A[δK−ρ + (1 − δ)L −ρ ] −1/ρ,

where A > 0, ρ ≠ 0, 0 ≤ δ ≤ 1, is concave for ρ ≥ −1 and convex for ρ ≤ −1.