Seminar 12

I give this number "12" to keep the order.  Tuesday the 4th of May, I hope to get it into the calendars. Also there is a workshop tomorrow the 29th.

This time the problems are not what I would call easy. There is an exam problem with a "worst of its kind" problem, and there are dynamic programs which are a stretch - but I think both are healthy to try. 

 

  • Quasiconcavity/quasiconvexity: consider the function LaTeX: f(x,y)=x+\sqrt{x^2-y}f(x,y)=x+x2y 
    • What do the level curves look like? 
    • Is the function quasiconcave? Quasiconvex? If not: on what domain(s) - if any - is it quasiconcave? Quasiconvex? 
       
  • Difference equation:

    Consider the following difference equation for the unknown v:
    LaTeX: p\cdot v_{t+1}-v_t+(1-p)v_{t-1} = Kpvt+1vt+(1p)vt1=K (that is on a form like Compendium 10-05 (b), you can handle that ...). 
    Here, p is a fixed probability, 0<p<1.  There is one value of p which is slightly quirky, hence the following formulation:

    • Find a particular solution of the form Ht for all p, except one. 
    • Find the general solution for all p in (0,1) except that one p.
    • Is it globally asymptotically stable for any p?

    (Problem taken from 3 (a) in this exam in ECON5150: http://www.uio.no/studier/emner/sv/oekonomi/ECON4140/v16/5150xh09.pdf Links to an external site. 
    The Department has discontinued Mathematics 4. If you think this application looks interesting, then try STK2130 Links to an external site. - but check first if it will count towards your grade, rules have changed.)


  • Dynamic programming: Exam 2008 problem 2. With a warning that you should think a little bit before you calculate too naively. 

  • Dynamic programming, I will give you one that has the limit transition T->infinity: Consider the problem LaTeX: J_{t_0}(x)=\displaystyle\max_{u_t\geq 0}\Big\{\beta^{T-t_0}(x_T+\sqrt{x_T})+\sum_{t=t_0}^{T-1}\beta^{t-t_0}\big(u_t+\sqrt{u_t}\big)\Big\}Jt0(x)=maxut0{βTt0(xT+xT)+T1t=t0βtt0(ut+ut)} where LaTeX: x_{t+1}=\frac1\beta(x_t-u_t)xt+1=1β(xtut) and LaTeX: x_{t_0}=x>0.xt0=x>0. The constant LaTeX: \betaβ  is in (0,1).
    • Calculate JT-1 and JT-2
    • By now you should be able to guess a form of the value function JT-τ(x) up to some Aτ that does not depend on x; prove this by induction. If you cannot guess, see cheat note a couple of items further down.
      • If you chose a nice form as per the cheat note, the difference equation for A should be "easy" to solve as well!
    • Let T go to infinity. State the Bellman equation for this problem.
    • Guess and verify a form for the value function. (Cheat note: this form is probably most convenient! Links to an external site.)
    • TOTALLY OPTIONAL: stochastic dynamic programming. 
      Go to the above 5150 exam, problem 1. In the dynamic programming equation, replace Jt+1(...) by the expected value LaTeX: \mathsf E[J_{t+1}((x_t-u_t)V_t)]=\frac12\cdot\big[J_{t+1}(0) + J_{t+1}(4(x_t-u_t))\big]E[Jt+1((xtut)Vt)]=12[Jt+1(0)+Jt+1(4(xtut))] .
      (Note: In theory, it is generally conditional expectation, conditioned upon what you know at time t. But in that problem, the shocks are iid.)
      Here is your task: Show that the dynamic programming equation for this problem, reduces to the one above!

  • The entire 2015 exam.
    It contains a "worst of its kind" problem. (With a video.)

 

 

By now, the following exams have been assigned: 

2003: problem 1 (sem#5), 2 (sem#6), 4 (sem#6) 
2008: problem 1 (sem#1 and #2 and #3) and 2 (now)
2009: the full set for sem#7 - the first full exam assigned!
2011: the full set for sem#10
2012: the full set for sem#8 
2015: now
2016: covered in full, split up over problem 1 (sem#3), 2 (sem#3 and 4 and 8), 3 (sem#10), 4 (sem#7)
2017: the full set for sem#9
2018: the full set for the virtual sem#11

As you see I have tried to leave full ones untouched.