Seminar problems (and a pre-seminar set too)

For May 3rd

For warm-up, you can do 9-01 and 9-03, although they will not be prioritized during the seminar.

• 9-07
• 9-12; here, show also that the solution is indeed optimal.
• 9-17
• The 2014 exam. 
  I suggest you simulate an exam situation to do this.
  I also suggest you bring a copy of the solution note to the seminar.

Earlier weeks' seminar problems follow below.  For reference, in case you want to catch up, I have as of yet (including the upcoming seminar) assigned the following exam and compendium problems.

Compendium problems:

• (1-01,) 1-03, 1-04, 1-05, 1-06, 1-07, 1-08, 1-12(a), 1-13, 1-14
• 2-01, 2-02, 2-03, 2-06, 2-07
• 3-03, 3-05, 3-07, 3-12, 3-13, 3-14, 3-21
• 4-01, 4-02, 4-05, 4-08, 4-09, 4-10
• 5-06, 5-14
• 6-10, 6-12, 6-15 (and generally, start from the beginning of section 6 to drill 2nd order diff.eq's)
• 7-01, 7-02, 7-04, 7-05
• 8-01, 8-06
• 9-02, 9-05
• 10-03, 10-04, 10-05
• 11-01, 11-04, 11-07

Exam problems

• Exam 2008 problem 1
• Exam 2011 problems 1d and 3

For seminar #1, February 2nd:

• Let f(x) = g(||x||). What is the gradient of f, in matrix form?
• Do compendium problems 2-01, 2-02, 2-03 and 2-07:
  • How much of these can you do without differentiating?
  • In 2-01 (b) and (d): write down the associated matrix, and work with that directly.
  • (Do the rest as well.)
• Use the determinant criteria to show that a Cobb-Douglas of three variables is strictly concave if the sum of exponents is <1.
• Exam 2011 problem 1 (d) (closing in on being able to pass an exam already, yay!)
• Prove from the definition of positive semidefiniteness that the odd powers \(\mathbf A^{2p+1}\)(p a natural number) of a positive semidefinite matrix \(\mathbf A\)is positive semidefinite.
  • Hint:  Put \(\mathbf y=\mathbf A\mathbf x\)
  • If \(\mathbf A\) is also positive definite, what about the odd powers?
  • From the determinant criteria: Could \(\mathbf A^3\) be positive definite if \(\mathbf A\) is merely positive semidefinite, but not positive definite? 
  • What about odd powers of negative definite matrices?
  • What about even powers?

For seminar #2, February 9th:

• Exam 2008 problem 1 (c) (can be done without anything from (a) and (b)).
• Compendium problem 1-13
• 2-06
• 3-03
• 3-05
• 3-21 (a bit demanding?)

For seminar #3, February 16th:

• Exam 2011 problem 3 (the rest)
• Give a one-line argument why the admissible set of a concave program, is convex. Does the same have to be true for a quasiconcave program?
• 3-07
• 3-12
• 3-13
• Consider the problem of minimizing |x|+|y| subject to \((x-a)^2+(y-b)^2\leq 1\), where a and b are nonnegative and \(a^2+b^2>1\). Use Kuhn--Tucker to decide for what a and b there is a solution at \((x,y)=(0,1)\).
• Possibly to be postponed: 3-14
• Not to be covered in this seminar - but maybe the next one: This problem. 
  (Those of you who have taken a course in the economics of information, may know it well already.)

For seminar #4, March 1st:

• Leftovers from last problem set (that would be 3-14 and the information economics problem?)
• 1-03
• 1-06
• 1-12 (a) (likely, the seminar will rather give priority to 1-06. If hard, try that one or 1-09 first)
• 1-07 ("mutually orthogonal" means that if you dot any pair, you get zero)
• 1-14
• Given a square matrix A: a non-null vector v is called an eigenvector for A if Av = ¦Ë v for some number ¦Ë. 
  Show that the matrix A - ¦Ë I cannot have full rank.

For seminar #5, March 15th:

• Exam 2011 problem 1. (You have already done 1d, now use a different method!)
• 1-04
• 1-05
• 1-08 (might be hard - if so, try 1-01 first)
• Here is an argument that an nxn  symmetric matrix has n linearly independent eigenvectors. 
  Addendum: You should not stress this problem too much - but you should remember the fact that a symmetric matrix has n linearly independent eigenvectors.
  I give the setup, and then three questions remain to complete the proof. Indeed, something stronger will be shown: such a matrix has n orthogonal eigenvectors.  
  • From class, you saw that the problem
    \(\max/\min \ \mathbf x^\top \mathbf A\mathbf x \quad\text{subject to}\quad \mathbf x^\top\mathbf x=1\) is solved by eigenvectors, so there is at least one.
  • Suppose for contradiction that there are only k<n linearly independent eigenvectors. Then there is a line through the origin, orthogonal to the span of these k eigenvectors.  
    • (For example, think a budget constraint with initial endowment z:  \(\mathbf p^\top(\mathbf x-\mathbf z)=0\) determines an \(n-1\)-dimensional hyperplane, and the nth direction is p and is orthogonal to all the possible net transactions x-z.)
  • It is possible to write the span of the eigenvectors as the span of orthogonal vectors \(\mathbf y_1,\dots,\mathbf y_k\). (For example, consider a plane through the origin; it is spanned by two vectors, but you can always find two orthogonal vectors (ninety-degree angle) that span the same plane.)
  • Use Lagrange's method to solve 
    \(\max/\min \ \tfrac12\mathbf x^\top \mathbf A\mathbf x \quad\text{subject to}\quad \tfrac12\mathbf x^\top\mathbf x=\tfrac12\ \text{and }\mathbf x^\top\mathbf y_i=0\) for all these \(\mathbf y_1,\dots,\mathbf y_k\). 
  • Lagrangian: \(L(\mathbf x)= \tfrac12\mathbf x^\top \mathbf A\mathbf x - \tfrac12(\mathbf x^\top\mathbf x-1)-\mathbf x^\top\sum_{i=1}^k \mu_i \mathbf y_i\) . Stationary if 
     \(\mathbf x^\top \mathbf A = \lambda \mathbf x^\top +\sum_{i=1}^k \mu_i \mathbf y_i^\top\) . 
  •  Right-multiply with some \(\mathbf y_i\):   \(\mathbf x^\top \mathbf A \mathbf y_j= \lambda \mathbf x^\top \mathbf y_j+\sum_{i=1}^k \mu_i \mathbf y_i^\top\mathbf y_j\) .
  • Question 1: Why is the left-hand side zero? (Hint: why is \(\mathbf A\mathbf y_j\)a linear combination of vectors orthogonal to \(\mathbf x\)?)
  • Question 2: Why does this prove that \(\mathbf \mu_j=0\)?
  • Question 3: Why does this prove that \(\mathbf x\) is yet another eigenvector, which - contrary to assumption - cannot be written in terms of the k ones we started with?

Update. More problems because you have another week, see Messages.

• The following problem consists in finding the eigenvectors of a 2x2 triangular but not diagonal matrix. It suffices to cover either upper or lower triangular matrices (the other one works the same way); by scaling the matrix, we can assume the off-diagonal element to be -1. Consider therefore the matrix \(\mathbf A = \begin{pmatrix}a&-1\\0&d\end{pmatrix}\)
  • Find all the eigenvalues for all values of a and d.
  • Is there any case which is "special" in any way compared to the problems you have seen before?
• 4-01 (maybe do 4-02 first?)
• 4-02
• 4-10

For seminar #6, March 31st:

• Compendium problem 4-05
• Compute the the double integral of xe^y (the same integrand as in 4-05!) over the domain bounded by y = 0, x = 1 and y = x - in both orders.
• Let  p>1>q>0 and let f be a nonnegative function defined on the bounded domain between the functions y = x^p and y = x^q. Formulate in terms of iterated integrals - both the dx dy order and the dy dx order - the volume under the graph of f. 
• 4-08 (trigonometric functions!)
• 4-09 (ditto!)
• 4-10
• Correction! Wednesday night, Yikai pointed out to me that I had reversed order here. Stupid mistake, now corrected. Let A be nxn, and let the columns of an nxn matrix V be eigenvectors of A. 
  • Show that AV = VD where D is diagonal and element ii is the eigenvalue associated with the ith column of V.
  • Suppose in the following that the columns of  V are linearly independent (that is possible if and only if A has n linearly independent eigenvectors), so that we can invert  V and write A = VDV^-1. (This is called diagonalization. The case where A has n linearly independent eigenvalues, makes lots of things easier.))
    Compute the matrix power A²⁰¹⁶ of such an A.
  • Compute the matrix exponential exp A = I + A + A²/2! + A³/3! + ... and the matrix exponential exp tA (where t is a real number).
  • Compute the matrix product exp(tA) exp(-tA).
• Matrix exponentials do not necessarily behave as nice as you are used to from e^x. For example, it is not universally true that exp(A+B) = exp(A) exp(B).
  Give a criterion that this formula holds. (It is way more general than when one of the matrices is a scaling of the other!)
• The eigenvectors in this course, can be called right eigenvectors, as opposed to a left eigenvector p', which is such that p'A = ¦Ëp'.
  Explain how we the tools of the course can be adapted to find left eigenvectors.
• One application where left eigenvectors are more common, is Markov chains (earlier taught in ECON5160): Let X_t be a random process where the probabilities of tomorrow's state, depends only on today's state (i.e. not on previous history). If we have n states,  let p_ij = the probability of being in state j tomorrow given that we are in state i today. The matrix P = (p_ij) then has the following properties: If the initial state X₀ is drawn from a distribution is p', (that is, element i is the probability that  X₀ = i) then the unconditional probability distribution of  X_t is the vector p'P^t (the superscript is power ^t). 
  • A stationary distribution is a p' one such that if X₀ is distributed  p' then all  X_t are (unconditionally distributed  p' as well. 
    Show that  p' is a left eigenvector of  P, and find the eigenvalue.
  • Some processes have a unique limiting distribution  ¦Ð such that as \(t\to+\infty\), p'P^t will converge to ¦Ð no matter what p'
    Show that such a  ¦Ð  must be stationary as well.

For April 7th:

As announced, the Tuesday will be a lecture. Thursday will be a lecture, and then possibly, if time permits, some problems will be covered.

• The problems to be given priority on Thursday, will be 6-15, 7-01 and possibly 6-10. (If you need "simple" 2nd order diff.eq. problems for a start, then start at 6-01, then 6-02 ...)
• Other problems you should do: 5-06 (at least the "derive" part), 5-14, and 6-12 (I have not yet covered equilibrium states: it is just a constant solution).

For April 12th:

• 7-02
• 7-04
• Exam 2008 problem 1. (You have already done part (c).)
• Classify the origin as equilibrium point for the system x' = 1 - exp(x-y),  y' = -y.
  (Classify only locally. This is an example from the book, under Olech's theorem for global asymptotic stability. That test is not curriculum, although it is not hard - try it!)
• 7-05

For April 19th

• It is a fact that a product of two odd integers, is an odd integer. Formulate as proof by induction the fact that the product of any (natural) number of odd integers, is an odd integer.
• Show by induction that 1 + 2 + ... + n = (n+1)n/2.
• Show by induction that for every natural number n, the number \(9^n-1\)is divisible by 8. (That is, (9ⁿ-1)/8 is an integer.)
  Hint: Write \(9^{n+1}=9^n\cdot (8+1)\)
• 10-03
• 10-04
• 10-05
• Look at the difference equation given in problem 3 (a) in this exam in ECON5150. Here, the unknown function is \(v\), while  \(p\in(0,1)\) and \(K\) are constants. (Do not be confused by the running index being called i rather than t - it is not supposed to be time in this application.)
  • Is there any \(p\in(0,1)\) such that the equation is stable?
  • Find the general solution.
  • Find the particular solution which satisfies \(v_0=v_N=0.\)
    (The Department has discontinued Mathematics 4. If you think this application looks interesting, then try STK2130 - be aware though, that 2*** courses may not be useful towards your degree even though the content is at a level we used to assign to PhD candidates.)

For April 26th

Dynamic programming: 

• 11-01
• 11-04
• 11-07
• This infinite-horizon problem: http://www.uio.no/studier/emner/sv/oekonomi/ECON4140/h13/dynprog_infinitehorizon.pdf  

Continuous time:

• 8-01
• 8-06
• 9-05
• 9-02