MATH2003J, OPTIMIZATION IN ECONOMICS,
BDIC 2023/2024, SPRING
Problem Sheet 14
Question 1:
Compute the principal minors, the Hessian, and the bordered Hessian of the following func tions:
(a) f(x, y, z) = e
x+y + z.
(b) g(x, y) = xy.
(c) h(x, y) = e
x
cos y.
Question 2:
Consider the function
defined for x, y > 0.
(a) Sketch the level sets Ca and the upper level sets Pa of f, for a = 1, 0,−1.
(b) Use the definition to examine if f is quasi-concave.
(c) Use the Second order derivative test(s) to examine if f is concave and/or quasi-concave.
Question 3:
Let f be a function of n variables defined over a convex set S ⊂ Rn.
(A) f is quasi-concave (i.e. the upper level sets Pa are convex for all a).
(B) f has the following property: “for every points x, x0 ∈ S which satisfy f(x) ≥ f(x0) then they also have to satisfy: f((1 − λ)x + λx0) ≥ f(x0) for all λ ∈ [0, 1] .”
Show that the above 2 statements are equivalent. That means (A) implies (B) and (B) implies (A).