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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

(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

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