ECON4140 - Autumn 2013

Seminar problems

This is a live document to be updated weekly. Hopefully I get to a more efficient editing of the course web page in early september. In the meantime, the problems at the 2012 course page can indicate which problems will be given.

For seminar #1:

  • Problem 1(d) from the 2011 exam.
  • Problem 3 from this set; this time I did do the concavity criteria yes. Furthermore, establish concavity without differentiating. (And, you can very well do the others too.)
  • From the exam problem collection, do problems 2-01 (and, what is the definiteness property in part (d)?), 2-02, 2-03, 2-06, and 2-07.

 

For seminar #2:

(This is same set as last year, only reproduced below. A bit mixed. Jensen's inequality I think I have not mentioned by name – look it up.)

  • Exam 2008 problem #1 part (c) (to be done without utilizing any results from (a), (b))
  • Exam 2009 problem #1 (a), (b)
  • From the compendium: 1-07 (recall the definition of orthogonality: dot product = 0)
  • From the compendium: 1-13
  • From the compendium: 2-04
  • Let g(y) = f(Ay), where y is an n-vector and A is a matrix of order m by n.
    • (a) Calculate the gradient and Hessian of g and express them as matrix products.  Be careful to get everything in the right order.
    • (b) Let ri T (the transpose of ri) be row number i of A, and let ci be column number i.  Which one of f(ri) or f(ci) is well-defined (no matter what m is)?
      Denote the well-defined one as zi and gather these in a vector z.
    • (c) Let h(y) = g(y) – zTy.  Calculate the gradient and Hessian of h.
    • (d) Suppose g is convex and that we are given the problem to maximize h subject to y being in a given compact (= closed and bounded) set S. Show that the maximum must be on the boundary.
    • (e) use part (d) to find the maximum of h in the case where n = 2, and S is the set of all y such that all yi ≥ 0 and their sum is ≤1.  
      (In case of character set issues: the ≥ symbol is supposed to be greater-than-or-equal.)

  • An application case:
    A risk manager of a savings bank attempts to explain the potential for losses on mortgages in a potentially averse market:
         «In our mortgage portfolio, the average loan-to-value ratio is 80 percent, so even if our customers were totally broke, apart from the house pledged as collateral – then we could stand a 20 percent price drop without any other losses than the administrative cost of collecting the collateral.»
    Q: What is wrong with this argument? Formalise using Jensen's inequality.

 

For seminar #3:

Same as last year:

  • 2005 #2
  • 2007 #3
  • 3-03
  • 3-05
  • 3-12
  • 3-14

If this is not enough, you might have a look at looking at http://www.uio.no/studier/emner/sv/oekonomi/ECON4120/h11/undervisningsmateriale/Wk38_inf_ec_problem.pdf

 

 

For seminar #4 (I didn't realize I had assigned a couple of these earlier ... my bad!):

Nonlinear programming problems:

Extra problem not to be covered: you should now be able to solve http://www.uio.no/studier/emner/sv/oekonomi/ECON4120/h11/undervisningsmateriale/Wk38_inf_ec_problem.pdf

Linear algebra problems:

  • 1-06
  • 1-07 ("orthogonal" means that the dot product is zero)
  • 1-13

 

 

For seminar #5:

Linear algebra:

  • Exam 2011 problem 1 - I have more or less covered in class, so it will likely not be covered, but you should definitely be able to do it.
  • 1-05
  • 1-08 (if you think it looks hard, try 1-01 first)
  • 1-10
  • 1-17
  • 1-18

Integration:

  • 4-01
  • 4-02
  • 4-10
  • 4-11

 

 

For seminar #6:

Double integrals:

  • 4-05
  • 4-06
  • 4-07
  • Compute the double integral of xey (the same integrand as in 4-05!) over the domain bounded by y = 0, x = 1 and y = x - both the order it stands and by reversing the order of integration.

Differential equations:

  • 5-04
  • 5-06
  • 5-11
  • 5-14
  • 6-08 (hint: integrate once first!)
  • 6-10 part (a).

 

 

 

For seminar #7:

These double integrals problems are included also to drill sin and cos:

  • 4-08
  • 4-09

Differential equations:

  • 6-02
  • 6-11
  • 6-15
  • 6-10 parts (b) and (c) (should not be hard by now!)
  • 6-13 with and without the suggested substitution in part (a).
  • 6-12 if time permits.
  • The "deduce" part of 7-01. (You should be able to do the rest as well, and it will most likely be assigned for seminar 8.)

 

 

For seminar #8:

  • The rest of 7-01
  • 7-04 (a) and (b). If you think (b) is hard, try 7-03 first, where you in part (b) put z=x+y and notice that z' can be written as h(z).
  •  Classify the origin as equilibrium point for the system x'=1-exp(x-y), y'=-y. (This is an example from the book, under Olech's theorem for global asymptotic stability. That test is beyond the course, although it is not hard.)
  • 7-05 (Whenever the problem says "draw", think "sketch". And, "integral curve" = path/orbit of a particular solution)
  • 7-06
  • I outlined the solution idea for exam 2008 problem 1. Do that one, except part (c) that has already been done. ((c) item 1 easily follows from (a) and (b). How?)
  • The remaining problems in this set from 2011.
    (The matrix exponential problem could very well be postponed in the interest of time.)

 

For seminar #9:

Difference Equations:

  • 10-03
  • 10-04
  • 10-05

Dynamic Programming:

  • 11-01
  • 11-03
  • 11-04

For seminar #10:

Dynamic Programming:

Calculus of Variations:

  • 8-01
  • 8-02
  • 8-03
  • 8-06

Control Theory:

  • 9-01

For seminar #11:

Control Theory:

  • 9-02
  • 9-03
  • 9-05
  • 9-07
  • 9-10
  • 9-17
Published Aug. 29, 2013 2:29 PM - Last modified May 20, 2016 2:20 PM
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