ECON 2050 SEMESTER 2, 2022 Assignment 2
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ECON 2050
SEMESTER 2, 2022
Assignment 2
Problem 1 (15 marks)
Let
f : R2 → R, (x, y) ›→ 3y4 + 6y2x − x3
Answer the following:
(a) Find all critical points of f . (6 marks)
(b) Classify each critical point of f as a local maximum, a local minimum, or a point that is neither a local maximum nor a local minimum. (6 marks)
(c) Does f have global extrema? If yes, determine all global maxima and all global minima of f . If no, explain why not. (3 marks)
Problem 2 (10 marks)
If g : Rn → R is convex and k ∈ R then S = {x ∈ Rn : g(x) ≤ k} is a convex set. Using this theorem, show that if a ∈ [−2, 2] and b ∈ R, then the following set, M, is convex.
M = {(x, y, z) ∈ R3 : x2 + y2 − axy ≤ 0, x + by + z ≥ 0}
Problem 3 (15 marks)
Solve the constrained maximization problem
max
(x,y,z)∈R3
yz + xz s.t. x2 + z2 = 1, yz = 4 (1)
using the Lagrange approach. You may assume that the constraint set is compact without proof. Explain each of your steps.
Problem 4 (60 marks)
Let c ∈ R+. Consider the maximization problem
max xy4 s.t. xey ≤ 3e2, c ≥ y, x ≥ 0, y ≥ 0. (2)
x,y∈R
For parts a and b below, assume that c = 0.
(a) Determine and briefly explain whether any points of the constraint set violate the non-degenerate constraint qualification of the Kuhn-Tucker theorem.
(b) Solve the maximization problem (2). (Hint: You do not need to apply the Kuhn-Tucker theorem nor any other theorem from the lecture slides.)
For the remaining parts of this question, assume that c > 0. For parts d and i only, further specifically assume that c = 2.
(c) Determine and prove whether the objective function f : R2 → R, (x, y) ›→ xy4 is concave, convex, both concave and convex, or neither concave nor convex.
(d) Draw the constraint set
C = {(x, y) ∈ R2 : xey ≤ 3e2, c ≥ y, x ≥ 0, y ≥ 0}
for the case c = 2. Using your figure, explain whether the constraint set C is convex (you do not need to formally prove (non-)convexity, but explain using your graph whether the constraint set satisfies the definition of convexity).
(e) State whether the maximization problem (2) is a concave program. Explain your answer.
(f) Does the Extreme Value Theorem apply to the maximization problem (2)? Explain why or why not, and if yes, also state what the Extreme Value Theo- rem implies for this problem.
(g) Determine and briefly explain whether any points of the constraint set violate the non-degenerate constraint qualification of the Kuhn-Tucker theorem.
(h) Solve this problem by applying the Kuhn-Tucker theorem. Make sure to de- rive all solutions to the first-order conditions, and carefully explain each of your steps.
(i) In a new figure, solve the maximization problem (2) graphically for the case c = 2. Illustrate the constraint set, some level curves of the objective function, and indicate in which direction the objective function increases.
(j) Let γ∗ denote the value associated with the solution to (2) of the Lagrange multiplier of the constraint c ≥ y. For which values of c is γ∗ zero? For every such c, is the constraint c ≥ y binding at the solution to (2)?
2022-10-13