Download Model of closed economy with rational expectations∗

Model of closed economy with
rational expectations∗
Miroslav Hloušek ∗∗
Faculty of Economics and Administration of Masaryk University in Brno,
Department of Applied Mathematic and Informatics, Lipová 41a, 602 00
Brno, email: [email protected]
Abstract
This paper shows the way of solving linear difference model
under rational expectations. The whole procedure is described in general form. The conditions for existence of a
unique solution are summarized. The system is solved first
for unstable part and then for stable part to get the final
solution. There is made an application for macroeconomic
model of closed economy. A reaction of agents in presence
of shocks, when anticipated or not, is examined. Finally
an influence of distinct types of Phillips curve on an adjusment process is discussed. The results are illustrated in the
figures.
Keywords
rational expactations, closed economy model, Phillips curve
1
Motivation
We have a model of closed economy with rational expectations. The model
is characterised by four equations that are expressed in log-linearized form:
(1)
yt = αyt−1 − βrt + ωt
(2)
πt = γπt−1 + (1 − γ)Et πt+1 + δyt + χt
(3)
rt = it − Et πt+1
(4)
it = κyt + φEt πt+1 + ξt ,
∗
This paper has been worked as a part of research activities at the grant project of
GA CR No. 402/02/0393.
∗∗
I thank Jaromı́r Beneš and Osvald Vašı́ček for useful comments and suggestions.
1
where yt is the output gap, rt is the real interest rate, it is the nominal interest rate, πt is the rate of inflation. All variables are expressed as a deviation
from their equlibrium value. Et at+1 denotes conditional expectations.1 The
parameters have these properites: α ∈ (0, 1), β > 0 , γ ∈ (0, 1), δ > 0 κ > 0
and φ > 1. Finally, ωt , χt and ξt represent exogenous shocks.
The equation (1) relates aggregate spending to lagged values of output
and the real interest rate; it corresponds to the aggregate demand equation.
The equation (2) is an inflation-adjustment equation in which current inflation depends on lagged and expected future inflation and output. This
equation (often referred to as Phillips curve or agregate supply) is based on
multiperiod overlapping nominal contracts. The equation (3) is an identity
for the real interest rate. The equation (4) is a monetary rule; the nominal
interest rate (instrument of monetary policy) depends on current output and
expected inflation. There occur three types of shocks: aggregate demand
shock, agregate supply shock and monetary policy shock.
2
Solving
The model’s equations will be put in the form:
(5)
AEt xt+1 = Bxt + C²t ,
where A, B are square matrices of coefficients that belong to vectors xt+1
and xt , C is matrix of coefficients of exogenous variables ²t . Et xt+1 =
E(xt+1 |Ωt ), where E(.) is mathematical expectation operator, Ωt is the information set at t. We suppose that all agents have the same information
at a given time.
For existence of a non-expolosive solution it is required that the exogenous variables ²t do not “explode too fast”. This conditon in effect rules out
exponential growth of the expectations of ²t .
Vector xt contains two types of variables: predetermined and non-predetermined. The difference between them is rather important. A predetermined variable is known at time t, a non-predetermined is not.2
If the matrix A is regular, the process of solving is simpler. But more
frequently, the matrix A is singular and therefore we use generalized Schur
form to solve it. For square matrices A and B we find matrices S, T , Q,
Z, where S and T are upper triangular matrices and Q and Z are unitary
1
2
Expectation of at+1 formed upon information available in period t.
The labelling of these variables is xpred.
and xunpred.
respectively.
t
t
2
matrices that satisfy
(6)
QAZ = S
QBZ = T.
We made a linear transformation of vector xt :
(7)
Z x̃t = xt ,
This transformation implies that all elements in vector x̃t contain information that has influence on an element in xt .
We substitute (7) into equation (5), premultiply by Q and use relations
(6) as stated above.3
(8)
AZ x̃t+1 = BZ x̃t + C²t ,
(9)
QAZ x̃t+1 = QBZ x̃t + QC²t ,
(10)
S x̃t+1 = T x̃t + D²t .
The ratio of diagonal elements of matrices T and S produces eigenvalues
of the system.
λ(A, B) = T (i, i)/S(i, i)
Blanchard-Kahn condition requires for existence of a unique solution,
that the expectation of xt does not explode and that the number of eigenvalues outside the unit circle is equal to the number of non-predetermined
variables.
We order the system by the value of eignevalues and write equation
(10) in well-arranged form. The part of vector x̃ that corresponds to stable
eigenvalues (in the unit circle) is labelled s, the unstable part is labelled u.
·
¸·
¸ ·
¸·
¸ ·
¸
S11 S12
st+1
T11 T12
st
D1
²t .
=
+
0 S22
ut+1
0 T22
ut
D2
2.1
Solving of unstable part
We write out the equations of unstable part for time into infinity and make
backward iteration. The unique stable solution for ut is then4
(11)
ut = −T22 −1
∞
X
−1
[T22
S22 ]k D2 ²t+k
k=0
3
4
For expectation value of x̃ we will use x̃t+1 instead of Et x̃t+1 for simplicity.
Here ²t+k represents expectation value.
3
2.2
Solving of stable part
We write the stable part in the form of difference equation:
(12)
(13)
S11 st+1 + S12 ut+1 = T11 st + T12 ut + D1 ²t
st+1 = S11 −1 (T11 st + T12 ut − S12 ut+1 + D1 ²t ).
We need to know the initial condition to solve it. We get it in the following way. The vector xt consists of predetermined and non-predetermined
variables and with regard to transformation Zt x̃t = xt we can write
·
¸ · ¸ " pred. #
st
xt
Z11 Z12
=
.
Z21 Z22
ut
xunpred.
t
For upper part
Z11 st + Z12 ut = xpred.
,
t
where ut has been solved and the value of xpred.
is known in time t. It is
t
easy to get
st = Z11 −1 (xpred.
− Z12 ut )
t
and substitute into (13) to get the whole trajectory of vector s. The final
solution results from transformation
xt = Z x̃t .
3
Application
Recall the model of economy from chapter 1. Now we can examine a reaction
of agents at presence of individual shocks.5 Due to shortage of space we pay
our attention only to aggregate demand shock. So the value of ωt in our
model is set ωt = 1, other variables are at their equilibrium level that is 0.
To enhance transparency we display only path of output and inflation rate.
The result can be seen in the Figure 1.
First, we assume the shock is unexpected. Output and inflation both
rise which is in accordance with economic intuition. The output returns
more sharply than inflation. Both variables come back to their equlibria
approximately after eleven periods. What happens when the shock is anticipated; it occurs e.g. in period 4. The agents expect rise of inflation in
5
For simulation we use these values of parameters: α = 0.8, β = 0.6, γ = 0.5, δ = 0.3,
κ = 0.5, φ = 1.5. They are set with reference to [3].
4
that period and therefore adjust contracts before the shock appears. The
inflation gradually rises from the beginning of our simulation. The response
of output is different. The gradual output decline is induced by the rise in
real interest rate (not shown here). The demand shock has direct impact on
output that rapidly rises in period 4. Both variables return to their initial
levels in a similar manner as in the previous case.
Unexpected shock
output gap
inflation
0.8
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
Periods
10
12
14
16
Expected shock
output gap
inflation
0.8
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
Periods
10
12
14
16
Figure 1: Aggregate demand shock – expected and unexpected
Now we examine the impact of distinct types of Phillips curve on adjustment process. The base value of parameter γ was 0.5; lagged and expected
inflation had the same influence. Figure 2 shows the response of output and
inflation to demand shock for different values of the parameter γ.
When γ = 0, the Phillips curve is forward-looking. Output and inflation
rise and then smoothly return to their initial levels that reach in period 7.
When γ = 1, the Phillips curve is backward-looking. The inflation little
overshoots, reaches its peak in period 2 and then gradually returns to the
equilibrium level. The response of output is more variable with decline below
5
the baseline. This shock displays a great deal of persistence. The variables
are back at their equlibrium in period 16.
Forward−looking Phillips curve
output gap
inflation
0.8
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
Periods
10
12
14
16
Backward−looking Phillips curve
output gap
inflation
0.8
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
Periods
10
12
14
16
Figure 2: Aggregate demand shock – different types of Phillips curve
References
[1] BLANCHARD, O.; KAHN, C. The Solution of Linear Difference Models under Rational Expectations, Econometrica, Volume 48, Issue 5
(Jul., 1980), 1305-1312
[2] KLEIN, P. Using the generalized Schur form to solve a multivariate
linear rational expectations model, Journal of Economic Dynamics and
Control, Volume 24, Issue 10, (Sept., 2000), 1405-1423
[3] WALSH, C. E. Monetary Theory and Policy. Cambridge: The MIT
Press, 1998. ISBN 0-262-23199-9
6