Saddles and Rational Expectations. Language barrier preventing me to solve this exercise

Question

For the system \begin{align} \dot{x} = x + y + 1\\ \dot{y} = 2x - y + 5 \end{align}

(i) Find the fixed point.

(ii) Transform the system into deviations from the fixed point. What are the characteristic roots of the transformed system?

(iii) Derive the equations for the stable and unstable arms.

(iv) Set the model up on a spreadsheet and plot, on the same graph, the following trajectories:

(a) The trajectory from initial point (-2,4)

(b) The unstable arm passing through the point at which $x=2$

4th exercise page 172 in the "An Introduction to Economic Dynamics book"

Can anyone explain to me how do we solve - (ii) Transform the system into deviations from the fixed point. What are the characteristic roots of this transformed system ?

I simply do not understand the point of the exercise. Anyone care to explain what it wants us to do?

I am a Slovakian student, and the exercises in English are getting me stuck. Any help would be appreciated.

Answer 1

$\newcommand{\vect}[1]{{\bf #1}}$

Note that you can write the system in the form

$$ \dot{\vect{x}} = A \vect{x} + \vect{b} $$

where

$$ \vect{x} = \left(\begin{array}{c} x \\ y\end{array}\right), ~~~~ A = \left(\begin{array}{cc} 1 & 1 \\ 2 & -1\end{array}\right), ~~~~~ \vect{b} = \left(\begin{array}{c} 1 \\ 5\end{array}\right) $$

A fixed point $\vect{x}^*$ is a point for which $A\vect{x}^* + \vect{b} = 0$, which in this case is

$$ \vect{x}^* = \left(\begin{array}{c} -2 \\ 1\end{array}\right) $$

Now define the variable

$$ \vect{z} = \vect{x} - \vect{x}^* $$

and note that

$$ \dot{\vect{z}} = \dot{\vect{x}} = A\vect{x} + \vect{b} = A\vect{x} - A\vect{x}^* = A(\vect{x} - \vect{x}^*) = A\vect{z} $$

That is,

$$ \dot{\vect{z}} = A \vect{z} $$

What you need to do now is to find the eigen-decomposition of $A$