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
f(x,y)=x+√x2−y
- 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:
p⋅vt+1−vt+(1−p)vt−1=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
Jt0(x)=maxut≥0{βT−t0(xT+√xT)+T−1∑t=t0βt−t0(ut+√ut)} where
xt+1=1β(xt−ut) and
xt0=x>0. The constant
β 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 valueE[Jt+1((xt−ut)Vt)]=12⋅[Jt+1(0)+Jt+1(4(xt−ut))] .
(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.