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Thursday, May 7, 2009

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?

7 comments:

bk drinkwater said...

I'll set to work.

Dismal Soyanz said...

is that system specified right? i've not come across a system where the current is dependent on the future value....

fibby said...

Yes. 90% of the models specified in Gali are of that form.

Dismal Soyanz said...

g. i should have a look at Gali then. another thing to read when i have the time. lol

fibby said...

Of course, if the future dependence really bugs you, you're free to move the shocks to the LHS and invert the matrix. But then the stability condition will flip too, *and* you'll have a shock vector with parameters in it, which is kind of a pain to work with.

Future dependence arises from forward-looking pricing behaviour in the Phillips curve, and forward-looking consumption decisions in the IS curve. It's a fact of life. More interesting models have a backward-looking term as well (sticky prices/habit formation).

fibby said...

... unless, of course, you were pointing out that I should have put an E for expectation in front of the forward-looking term. In which case you're right, but it doesn't change the model.

bk drinkwater said...

OK, I've finally had a couple of hours to look at this properly.

It appears to be above my payscale :(

By that, I mean I can't find a simple linear-algebraic derivation of the conditions given.

BTW... the form of the matrix is a little simpler than preented here:

A=
[ X, Y ]
[ kX, kY+B]

where X = 1- (phi_y / sigma)
and Y = (1-phi_pi)/sigma

I found that marginally easier to work with.

I suspect you need to invoke some of the various theorems about spectrums.

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