Question B.3 Consider the minimization problem M(p, y) = min x U(x) s.t. p1 · x1 + ... + pn · xn ≤ y where U : Rn → R is continuous. Prove that the function M(p, y) : Rn + × R+ → R is quasi-concave. [Hint: the subscript + means that all elements of a vector are non negative and at least one is strictly larger than zero.]

Question B.3 Consider the minimization problem M(p, y) = min x U(x) s.t. p1 · x1 + ... + pn · xn ≤ y where U : Rn → R is continuous. Prove that the function M(p, y) : Rn + × R+ → R is quasi-concave. [Hint: the subscript + means that all elements of a vector are non negative and at least one is strictly larger than zero.]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Question B.3

Consider the minimization problem

M(p, y) = min x U(x)

s.t. p1 · x1 + ... + pn · xn ≤ y

where U : Rn

→ R is continuous.

Prove that the function M(p, y) : Rn

+ × R+ → R is quasi-concave.

[Hint: the subscript + means that all elements of a vector are non

negative and at least one is strictly larger than zero.]

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