6. Determinacy of the Rational Expectations Equilibrium
In consequence, we obtain the subsequent matrix form for the system of equations:
Ay = Bx + u,
where the vectors of unknowns y, x and the vector of disturbance terms u read as follows:
(62)
|
y:= |
^ xt πt,H πt,F ʌ |
, x := |
' Et [xt +1 ] ■ Et[ x*+1] |
, u := |
'—Et [∆ at+1Γ Ut * 0 |
|
it-1 |
it |
ω1 vt | |||
|
ʌ Lh-1J |
ʌ L i* J |
L -(ω* )-1 v* J |
The coefficient matrices A, B, however, read:
|
1 |
0 |
0 |
00 |
0 |
0 | ||
|
0 |
1 |
0 |
00 |
0 |
0 | ||
|
-μ |
0 |
1 |
00 |
0 |
0 | ||
|
0 |
* -μ |
0 |
10 |
0 |
0 |
, | |
|
0 |
0 |
1 |
-1 1 |
-1 |
1 | ||
|
0 |
0 |
α ω |
00 |
1 |
0 | ||
|
0 |
ι* ω* |
0 |
_ * * 0 ω* |
0 |
1 | ||
|
1 |
0 |
ρ-1 |
0 |
■& |
-ρ-1 |
0 | |
|
0 |
1 |
0 |
ρ- |
1 μ* |
0 |
_ |
ρ-1 |
|
0 |
0 |
β |
0 |
0 |
0 |
0 | |
|
B := |
0 |
0 |
0 |
β |
0 |
0 |
0 |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 | |
|
0 |
0 |
0 |
0 |
0 |
ω-1 |
0 | |
|
0 |
0 |
0 |
0 |
0 |
0 |
( ω* )-1 |
In order to determine the eigenvalues of the system of equations (62), it has to be rearranged in the
following form:
y = Mx + v, (63)
where M := A-1B and v := A- 1u. Moreover, A-1 denotes the inverse of A, which exists because
det(A) = 1 = 0.
The matrices A-1 and M and the vector v read as follows:
18