ECON8013: Optimization for Economics and Financial Economics Assignment 2 Semester 1 2022
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ECON8013: Optimization for Economics and Financial Economics
Assignment 2 Semester 1 2022
1. - [22 points] Consider the problem
max x2 + y2 + y + 1
y2 +x2 -1S0
(a) [2points] Is there a global solution? Apply speciﬁc theorems from the course
to explain your answer.
(b) [2points] Formulate the Lagrangian.
(c) [3points] Apply the appropriate cook book procedure.
(d) [3points] There should be 3 candidate solutions from (c). Identify them. (e) [2points] Which point from (d) does not satisfy the constraint qualiﬁcation?
(f) [2points] Illustrate via a diagram the global maximum, where gradients, and
the constraint set appear.
(g) [7points] Find a further constraint g2 : R2 → R that changes the optimal
solution. Repeat step (b), (c) and (f).
(h) [2points] If the max becomes a min in the optimization, what further infor-
mation can you say about the optimum?
2. . [18points] Consider the following constrained Maximization problem
max x + y
subject to (x · 4)2 + y < 7 and (x · 5)3 · 12y < ·60.
(a) [3points] Does a global solution exist? Is it unique. Argue as in 1(a). (b) [3points] Illustrate via a diagram the constraint set.
(c) [4points] Specify the KKT necessary optimality conditions, with comple- mentary slackness conditions
(d) [8points] Solve the problem