I am a university student specializing mathematics for economists. I am in a preparation for my final exam. My prof gave me some questions that might be on the exam. One of the question dragged me down so hard. I cannot even imagine where to start. If you guys have any idea about this, please help me. If I prepare this, I feel like I will do well on the exam. Thanks a bunch in advance.
The question is,
Let m(α,y) be defined as the minimum value of αx subject to g(x) >= y, where α, x∈ R^n++, y∈ R+, and g(x) is strictly monotonic increasing and quasi-concave. Prove that m(α,y) is (i) non-decreasing in α and y and (ii) concave in α. Then, given that g(x) is homogeneous of degree k, derive the corresponding form of m(α,y)