[SOLVED] MATH2003J OPTIMIZATION IN ECONOMICS BDIC 2023/2024 SPRING Problem Sheet 13 C/C

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.