The only time he's satisfied

The Favourable Bet Theorem is a statement about the tradeoff between money and risk. It says that a rational agent will prefer some risk over total certainty if the risk offers a chance of getting a higher payoff, but only if the agent is allowed to choose how much money to sink into the risky venture. You might not bet $100,000 even at very good odds (like 10:1), but given the choice, most people would bet at least $1 at those odds.

The current Micro assignment, due Thursday morning, asks us to apply the Favourable Bet Theorem to an actual student health poster about the risky business of having sex while under the influence of alcohol.

I do not know where to go with this one. You can choose a little bit of a bet, but you just can't have a little bit of sex.

Yes, I know House of the Rising Sun is not originally Bob Dylan's, but with a song so frequently covered I reserve the right to reference whichever version I want to.

Surely makes you lose your mind

This intriguing problem has come up in Macro.

We're analysing dynamic systems (on a very, very basic level) and working out when they are "determinate" and/or "stable". Before anyone brings in the heavy artillery, I should say we haven't used any actual differential equation theory. We've been told this: if \epsilon represents a stochastic exogenous shock (with mean zero), and the system is

 \left[ \begin{array}{c} y_{t} \\ \pi_{t} \end{array} \right] = A \left[ \begin{array}{c} y_{t+1} \\ \pi_{t+1} \end{array} \right] + \left[ \begin{array}{c} \epsilon_{y,t} \\ \epsilon_{\pi,t} \end{array} \right]

then the system is "determinate" if and only if both the eigenvalues of the matrix A are inside the unit circle. "Determinate" seems to mean having a unique solution (economists are pussies), and is taken to be synonymous with "stable" (lazy pussies, at that).

On occasion we work with a model where the form of the matrix is simply heinous, and calculating the eigenvalues becomes difficult. In case of this, clever textbook writers have come up with conditions which imply that both eigenvalues will be inside the unit circle. However, they don't always state the assumptions necessary for the implication to hold.

The problem with the current Macro assignment is that a question was designed using some of these clever stability conditions, but that those particular conditions rely on unwritten assumptions which the assignment question violates, with the result that the system of the assignment is actually not stable at all.

The matrix of the system has the form:

A = \left[ \begin{array}{cc} 1-\frac{\phi_{y}}{\sigma} & \frac{1-\phi_{\pi}}{\sigma} \\
\kappa(1-\frac{\phi_{y}}{\sigma}) & \beta - \frac{\kappa}{\sigma}(\phi_{\pi}-1) \end{array} \right]

The conditions which supposedly imply eigenvalues inside the unit circle (Gali 2008, pg 79) are:

\kappa (\phi_{\pi}-1) + (1-\beta)\phi_{y} > 0

\kappa (\phi_{\pi}-1) + (1+\beta)\phi_{y} < 2\sigma(1+\beta)

These conditions fail when \sigma=\beta=\kappa=1 (EDIT: and \phi_{y}=0). Why? What are the unwritten assumptions under which the conditions can be derived, and which of the parameter values are violating them?

In a cardboard box

Anyone else want to have a go at this? I'd really appreciate your comments. Blaise? Bruce? qwandor?

Stone you when you're trying to go home

I was going to look for a photo of a fresh egg and a fried egg, and be all like "this is your brain at 9am" "this is your brain at 4pm". But my brain was too fried to do it.

Micro is finished. Four days early.

Metrics is finished. It includes a long section where I explain why the proof in the lecture notes is wrong, then admit that I can't fix it myself. Due tomorrow.

Macro isn't started. Three days to go.

I left my seat twice today, for less than five minutes each time.

I was confused

Hi Dr Micro,

I'm having some trouble with question 3(b)... my solution has 16 distinct utility functions. Am I on the wrong track?

Thanks,

Fibby

Sometime tomorrow morning, Dr Micro's office: "Sixteen utility functions? то ужасно! OF COURSE YOU'RE ON THE WRONG TRACK YOU STUPID GIRL"

But I cannot figure out which direction I should go to correct myself. Halp!

... upon which halp arrives, in the form of an instant reply from Dr Micro. Awsum :D

An exit lights the sky

No math tutorial today. It was confusing, and it also left me with no reason to go up to Kelburn. I truly loathe the Pipitea campus. I loathe its interior architecture, its shiny new soulless feel, its lack of clearly defined study spaces, its noise, its filth, its shortage of sofas at lunchtimes, its little low chairs and computer desks with insufficient legroom, and its commerce students [especially the way they fail to respect others' study, or others' study room bookings]. I don't like being a commerce student when it requires me to study in such conditions. It makes me want to scream.

Assignment 7 is about half done. At last I begin to understand the phase portrait thing! I can even sketch them by hand now, most of the time [but still haven't figured out how to plot them in Maple]. With understanding comes a gradual respect for differential equations, which is a relief. As I keep telling myself, I love mathematics - but it's better when I don't need to keep telling myself so.

This afternoon I got so engrossed in math that I nearly forgot to go to a lecture. It was a great feeling. Just for those few minutes of the day, it didn't matter that I was at Pipitea.

Big noise playing in the street

Gaah, I've spent the entire day on the same "mechanical" question. I should heed my own advice more often - labelling a problem "mechanical" is akin to calling it "easy", which I swore never to do again.

My first two attacks on this problem resulted in incorrect eigenvalues after I kept making silly errors while simplifying the characteristic polynomial. After I'd finally figured out that there was only one eigenvalue, I then tried to be clever in finding the invertible matrix Q with the property that A=QBQ^-1. Cleverness got me nowhere fast, and after an hour of abortive attempts and checking, I finally had to slow down and come at the thing systematically. It was only then that I realised I had an insoluble system on my hands. After a bit more checking of every stage of my workings I arrived at the conclusion that something funny was going on. The only thing left to do was flick a despairing email to Chris, which I did as quickly as possible.

Chris, being a terrific lecturer and extremely available even on weekend afternoons, replied promptly with a small novel's worth of explanation and elaboration. I've worked over it a bit, but more work is in order. And those two proofs still need to be done. And I haven't written up yet. And the assignment is due in 17 hours. Suddenly I'm glad I already got so far ahead on the test...

Mona Lisa just keeps on smiling

I haven't actually done any math since Friday. Waiting to do some before writing about it doesn't seem to be getting me anywhere. Instead, I intend for this post to shock me back into action - not an easy trick after such a bad loss of momentum. If I can get angry enough with my own lassitude then I'll be fine.

I hereby declare that I am angry and will do some math now.

Never could get the hang of Thursdays

Yea, the Avenging Analyst did battle with the assignment-beast from the third hour to the sixth hour, despite her trusty fellow warrior being spaced out on medication; and though the beast bled from many wounds, verily, it was not slain. The Fearless Functional required of the Master additional information, and the Master did produce a rare and shining Weapon, the true definition of the determinant, which should have been taught in MATH 207 but was not. The Analyst was struck with fear, for she was but a Student, and the Weapon was only to be wielded by those who were true and pure of heart, and also knew what a permutation inversion was. The Miserable Mathematician did wonder if the last part of the assignment-beast was not now rendered impossible; but as the rest was More Or Less Done, she resolved not to despair, but to stand strong in the face of adversity, and strive to wield the Weapon herself.

But verily, there was no opportunity to examine the Weapon more closely, for it was then time for the Lecture. During the Lecture, the Master made various amusing statements, viz "if these were easy, we would have them in an economics course" and "you can construct a universe nicer than ours in which [infinitesimals] have a genuine meaning". In addition, the Analyst did gain a grade of 16/20 for Assignment 3, which caused her great joy. These moments of brightness brought the Analyst relief of mind, but no sooner had she gathered strength to battle again with the assignment-beast [refreshed by having seen much matrix algebra in the Tutorial] than it was time to journey to the loathsome Pipitea Campus and partake in Other Activities. It became clear to the Analyst that her foe would be left unmolested until after nightfall, and she despaired of vanquishing it before the Deadline.

It must be a Thursday, said the Analyst.

One thousand and one yellow daffodils

A long day of staring at my notes, feeling stupid, tired, coldy and demotivated. Things finally started to go click about an hour ago. Assignment 4 deals with implicit functions over the space of n × (k+l) matrices - not something I've ever worked with before. I have that strange, half-numb, groping feeling that comes from working with formulae which describe something I don't yet understand.

Question 1 of the assignment asks us to prove that there is a unique z for every x in the relation z⁷ +x⁴z = x⁵ + 1. I did this using what Chris has called the Implicit Function Theorem, second version [seems to be similar to the Wolfram MathWorld version]. This is probably an example of what, in first year, we used to call "using a steam roller to crush a walnut", but I don't really know how else to go about it. I don't like this practice of having to start the assignment before the first tutorial.

Correction: I didn't get 1.5/2 for question 3 of assignment 2 - it was actually 0.5/2. Today I forced myself to go through the model answers for the assignment, a somewhat informative exercise.