ECON6003/6703 Mathematical Methods for Economics Final Exam, S1 2022

Instructions

1. Exam Duration:  2 hours and 30 minutes (150 minutes) This includes a sug- gested 10 min reading time at the start, but you may begin writing the exam any time.

Submission Deadline. Check Exam Canvas site for details of additional grace period for exam submission.

2. Exam  consists of Problem 1, Problem 2 and Problem 3, each with sub-parts. Hand- written answers must be combined into one PDF file and uploaded via Canvas Assign- ments link.

3. Points for each Problem/question are indicated against them. Your score in this exam is worth 45% of your final grade.

4. At various places you are asked to “Explain ...” something or other.  Please do not write essays . Only a short logical argument, or a proof is expected.

Problem 1                 (20 pts)

This question has five sub-parts. The answer to each part can be obtained by an argument involving a couple of lines or simply stating an appropriate result covered in our classes. Keep your answers brief — about 4 lines max.

1) Consider hyperplane H (p; 0), where the notation is as used in class.   What is the angle between any x 2H(p; 0) and p? Explain your answer.                                    (4 pts)

2) Consider the following three vectors

    A(1)   and     A(1)

What is the value of a for which these vectors are linearly dependent .           (4 pts)

3) The figure below shows a bounded region C and a point P in R2 .  The Separating Hyperplane theorem does not apply, and hence P cannot be separated from C . Com- ment.                                                                                                                  (4 pts)

P

Figure 1.

4) Upon setting up the Lagrangian, denoted by L, for the standard utility maximization problem of a consumer for two goods X and Y, facing a given set prices and income lead to the following first order conditions:

= y ¡ 入 = 0             = x ¡4入 = 0

@(@)入(L)  =  ¡(x+4 y ¡16) = 0

where 入 is the Lagrangian multiplier.

Use the above to deduce  the change in utility of the consumer from a marginal change

in her income. Which theorem did you use to get your answer?                        (4 pts)

5) Consider constrained maximization of F : Rn ¡! R  subject to constraints hj(北) = 0 where hj: Rn ¡!R, for j = 1; :::; m and F are twice continuously differentiable and as 入1 ; :::; 入m denote the corresponding multipliers as usual.

Let   (北  ; 入  ; :::; 入) be a critical point of the Lagrangian and consider the following statements

a) V = {rh1 (北  ); :::; rhm(北  )g   is a linearly independent set.  b)  The set V is a linearly dependent set.  c) rF(北  ) = 0.

Which of the three statements, if true, will prevent us from concluding that  北  a local

maximum (even if the second order conditions are satisfied?)                          (4 pts)

Problem 2                                                                                   (25 pts)

An investor can invest in assets X , Y and Z .   Suppose  fractions x ; y and z of his total wealth is invested in these respectively, his average return is xux + y uy + z uz where ux; uy  and uz  are the respective mean returns on these assets.  It is also known that the corresponding variance of this portfolio is given by

V (x ; y ; z)  =  400x2 + 400z2 + 200xz + 1600 y2 + 400 y z :

1. Suppose the investor is wanting to find a portfolio that achieves a targeted return of R but has the lowest variance/risk. Explain why he must be solving the following problem: (6 pts)

min V (x ; y ; z)  subject   to  x + y + z = 1

x ; y ; z

xux + yuy +zuz = R

2. Suppose ux = 1; uy = :15, uz = :1 and R = :12. Using the Lagrangian method, solve the above problem for the variance/risk minimizing portfolio.                                (15 pts)

Hint: You should be able to use a couple of constraints to solve out for z . You should then be able to solve out for x ; z by looking at their corresponding partial derivatives.

3. Using the Lagrange multipliers,  calculate the approximate change in risk if the above portfolio were changed optimally to obtain a target of 12.5% return instead of the 12% return above.                                                                                                      (4 pts)

Note: If for some reason you are unable to solve out for the multipliers in the previous part, write down the procedure.

Problem 3         (25 pts)