Real Analysis Suppose the utility maximization problem is as follows: Max (x,y) U = xy1/2 with constraints B: x + 2y ≤ 40 and x + y ≤ 25 and x,y ≥ 0 a. Describe and verify whether budget set B is open? Closed? Bounded? Compact? (Hint: You can use graphic help) b. There are two points a = (20,0) and a' = (0,4), where a,a' ∈ B, if λ = 1/2 . find the coordinates a = λa + (1-λ)a' ! (Hint: at a(x̄ ,y̅) -> x̄ = λx + (1-λ)x' and y̅ = λy + (1-λ)y') ! is a ∈ B (a on budget set)? c. Is B a convex set / not? Is B a strictly convex set? (Hint: You can use graphic help)
Suppose the utility maximization problem is as follows:
Max (x,y) U = xy1/2 with constraints B: x + 2y ≤ 40 and x + y ≤ 25 and x,y ≥ 0
a. Describe and verify whether budget set B is open? Closed? Bounded? Compact? (Hint: You can use graphic help)
b. There are two points a = (20,0) and a' = (0,4), where a,a' ∈ B, if λ = 1/2 . find the coordinates a = λa + (1-λ)a' ! (Hint: at a(x̄ ,y̅) -> x̄ = λx + (1-λ)x' and y̅ = λy + (1-λ)y') ! is a ∈ B (a on budget set)?
c. Is B a convex set / not? Is B a strictly convex set? (Hint: You can use graphic help)
d. Is the utility function in the maximization problem in the form of concave / convex?
e. According to Weierstrass's theory, do solutions to the utility maximization problem exist? Explain!
*if you cannot answer to all the questions above, please just answer to the d and e thankyou.*
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