Closed economy macro dynamics: AD-AS model
and RBC model.
Ragnar Nymoen
Department of Economics, UiO
22 September 2009
ECON 3410/4410: Lecture 6
Lecture notes on closed economy macro dynamics
AD-AS model
In‡ation targeting regime.
With reference to an earlier lecture, we know that the money
targeting regime will be qualitatively similar.
We will return to that regime in the open-econony part of the
lectures, because there will be qualitatuve di¤erences between
m-targeting and -targeting in the open economy.
Static and adaptive expectations— compare with rational
expectations in lecture 7.
The RBC model
IAM: ch 19
IDM: Ch 2.8.4
ECON 3410/4410: Lecture 6
Dynamic analysis; inf-target & static expectations
Stability of solution
We start from equation (20) in part II of slide set to closed
economy AD-AS model:
(yt
y ) = (yt
1
y ) + (zt
zt
1)
with:
=
=
Since both
and
2h
(1 + 2 b)
1
+1
are positive (as long as h > 0):
0 < 1=(1 +
)<1
so we conclude that the solution of (1) is (dynamically
asymptotically) stable.
ECON 3410/4410: Lecture 6
st
(1)
Deterministic solution
0. 05
5 percent above equilibrium initally
0. 04
Parameter
values set to
“…t” annual data
Parameter values from IAM p 565 and p IAM 569
α = 0 .8 7 8
γ = 0 .2
0. 03
See graph for
references to
IAM
0. 02
0. 01
1900
1920
1940
1960
1980
2000
2020
2040
ECON 3410/4410: Lecture 6
Stochastic solution
y-ga p (b ot h sho c k s)
y -ga p (o nl y d e m a nd sho c k )
0.025
and as in
the previous
graph
0.000
-0.025
1900
z
1920
1940
1960
1980
2000
2020
2040
zt =
0:7zt
s
1
+ "z t
st = 0:3st
0. 02
"zt
0. 00
1 +"st
IN(0; 0:01)
"st
IN(0; 0:005)
-0.02
1900
1920
1940
1960
1980
2000
2020
2040
ECON 3410/4410: Lecture 6
Dynamic analysis I
Dynamic multipliers and impulse-responses
Assume a steady-state situation initially, in period t 1, with
st 1 = vt 1 = 0: Then vt = 1 but vt+1 = vt+2 ::all equal to
zero. Hence we have a temporary demand shock.
Using (1):
@yt
1
=
@vt
1+
@yt+1
1
=
@vt
1+
@yt+2
1
=
@vt
1+
Set
@zt
@vt
@zt
>0
@vt
@yt
1
@vt
1+
@yt+1
@vt
@zt
1
=
@vt
1+
(
1
1+
1)
=1
The …rst multiplier is positive, but less than one, since the
interest rate is increased in the period of the shock.
ECON 3410/4410: Lecture 6
@zt
<0
@vt
Dynamic analysis II
Dynamic multipliers and impulse-responses
The second multiplier is negative, since SRAS has shifted up
in the same period as SRAD shifts back to its original
position. All the subsequent multipliers, or impulse responses
as they are called in IAM, are also negative, but they are
diminishing in magnitude.
See …gure 19.8 in IAM, which you can try to replicate with
the aid of the Excel programs that you have from the Lecture
plan page and/or seminars.
In‡ation responds more smoothly to a temporary demand shock.
This due to two features of the model.
1
Demand shocks only a¤ect in‡ation indirectly, through yt .
2
Movements in yt are smoothed by in‡ation expectations,
which weights heavily in in‡ation dynamics. See for example
…gure 19.6 and 19.8 in IAM.
ECON 3410/4410: Lecture 6
Temporary demand shock: graphical analysis
π
t
LRAS
AD
π1
π2
π
*
SRAS2
Initial
equilibrium in A
S R A S0
Temporary
shock in period
1 gives
equilibrium B
B
C
A
Equilibrium in
period 3 is in C.
y2 y0
y1
yt
ECON 3410/4410: Lecture 6
Long-run is back
in A
Inf-target & static expectations:permanent shocks I
In the case of a permanent demand shock, all the multipliers
are positive but declining:
0
=
1
1+
1
=
1
1+
j
=
1
1+
+
1
1+
(
1
1+
1) =
1
1+
2
j +1
! 0
j !1
Besides, we know, from the interpretation of the long-run
model, that the long-run multiplier “has” to be zero. The
algebra is seen to con…rm this. But what happens “in the
model”?
ECON 3410/4410: Lecture 6
Inf-target & static expectations:permanent shocks II
Note …rst that due to stability, we know that the system
reaches a new stationary equilibrium also after a permanent
shock, so we “only” need to …nd out what that new
steady-state is.
Go back to the system of equations:
yt
y
rt
=
(gt
= it
it
e
t+j
=
t
=
Mt
Pt
1
= r+
g)
e
t+1 ;
e
t+1 +
t+j 1 , for
e
t + (yt
= e m 0 Ytm 1 e
2
(rt
r ) + vt ;
(2)
(3)
h(
t
) + b (yt
y) ;
(4)
j = 0; 1;
(5)
y ) + st :
(6)
m 2 it
:
and note the these equation also hold in steady state.
ECON 3410/4410: Lecture 6
(7)
Inf-target & static expectations:permanent shocks III
Then, impose the long-run equilibrium conditions
yt = y
and
t
=
=
.
Finally, de…ne
v =0
since a permanent demand shock must be due to g , and
s = s, s T 0
for the possibility of a permanent change in s in the AS
schedule.
ECON 3410/4410: Lecture 6
Inf-target & static expectations:permanent shocks IV
y
y
=
y
= y
1
(g
1
g)
2
(r
r ) , AD
s, AS
(8)
(9)
Think of the y , g and r in (8) and (9) as the long-run values
that apply before a permanent shock shock. Explicitly:
y
y0 =
y
1
= y0
(g
g0 )
1
2
(r
r0 ) , and
s
(10)
(11)
with r and y as the endogenous variables.
ECON 3410/4410: Lecture 6
Inf-target & static expectations:permanent shocks V
Permanent demand shock:
@r
=
@g
1
,and
2
@y
=0
@g
Since there is no increase in the supply side determined
natural-rate of unemployment, the real interest rate must increase
to o¤set the permanently higher public real expenditure. This is
the economics behind the mathematical result we had above: that
the cumulated sum of the dynamic multipliers is zero.
Permanent (positive) supply shock
@y
1
= > 0 and
@s
@r
=
@s
1
<0
2
Since the natural (non-accelerating) level of now has increased, the
real interest rates comes down in order to increase demand.
ECON 3410/4410: Lecture 6
Adaptive expectations I
Which properties of the model are dictated by the choice of
static in‡ation expectations in the basic version of the closed
economy AD-AS model?
In order to investigate, consider adaptive expectations instead.
In the short-run model, replace
e
t
e
t 1
= (1
)(
e
t
t 1
=
t 1
by
e
t 1 ); 0
1;
(12)
or, equivalently:
e
t
=
e
t 1
+ (1
)
t 1:
(13)
Note …rst that the SRAD is una¤ected since in the Taylor rule
has been speci…ed in such at way that et+1 cancels out when
we substitute to derive the SRAD function.
ECON 3410/4410: Lecture 6
Adaptive expectations II
With adaptive expectations, the SRAS function
t
replaces
t
=
=
e
t 1
t 1
+ (1
+ (yt
)
t 1
+ (yt
y ) + st
(14)
y ) + st .
Short-run model.
Since the SRAD function is unchanged, and SRAS is given by
(14), we conclude that the short-run model is una¤ected by
changing the model of expectations.
The impact multipliers are therefore una¤ected.
Long-run model
It is also the same as with static expectations.
This is typical for models with expectations: the long-run
model is always una¤ected by changes in the speci…cation of
how expectations are formed.
ECON 3410/4410: Lecture 6
Adaptive expectations III
The dynamic analysis
This is a¤ected, since et 1 enters into the model, not only
t 1 as in the case with static expectations.
Intuitively however, dynamic stability is not endangered.
This is because, from (13) wr have:
e
t 1
= (1
)
1
X
j
t 1 j
j =0
which is a generalization of static expectations.
IAM p. 578 and 579 contain the detailed analysis, with the
…nal equations for yt y in (41) and for t in (42).
ECON 3410/4410: Lecture 6
A simple RBC model I
Macro production function:
Yt = Kt (At Nt )1
, 0<
< 1;
(15)
where Yt is GDP in period t, Kt denotes the capital stock, and Nt
is population in period t. At is the “technology parameter”, which
is however not a constant in this model:
Model for technological progress
ln At = gA t + as ;t ,
0 < gA < 1;
(16)
where the random part as ;t is given by the autoregressive equation:
as ;t = as ;t
1
+ "a;t ,
0
<1
where "a;t is an unpredictable technology shock.
ECON 3410/4410: Lecture 6
(17)
A simple RBC model II
An essential assumption is that saving is proportional to
income:
St = sYt ,
0 s<1
(18)
where s is the constant saving rate (this is modi…ed in
advanced RBC models).
As a simpli…cation, we assume that capital equipment lasts for
only one period, so that
Kt = St
(19)
1:
It is the infallible mark of RBC models that the labour marked
is assumed to be in equilibrium in each period. Labour supply,
NtS , is a function of the relative wage wt =wt :
NtS = N(
wt
);
wt
> 0.
ECON 3410/4410: Lecture 6
(20)
A simple RBC model III
wt is the steady state wage and
elasticity.
is the labour supply
When the wage is equal to the steady state wage, labour
supply is also equal to its steady state value.
Labour demand is obtained by assuming optimizing behaviour
by a macro producer, meaning that the wage rate is equal to
the marginal product of labour in each time period.
As shown in IDM, ch 2.8.4 (or in IAM ch 19.4) we have the
following …nal equation for the log output gap:
yt
y=
(1 + )
(yt
1+
1
y) +
(1
)(1 + )
as ;t :
1+
(21)
Hence, we have the important result that also in this
equilibrium model, there will be periods of booms (positive
output-gap), and troughs (negative output-gap)
ECON 3410/4410: Lecture 6
Simulated solution of the RBC model
0.03
0.02
= 0:1,
0.01
= 4;
0.00
= 0:5
"a;t
-0.0 1
-0.0 2
190 0
191 0
192 0
193 0
194 0
195 0
196 0
197 0
198 0
199 0
200 0
201 0
ECON 3410/4410: Lecture 6
N(0; 0:01)
RBC and AD-AS model I
In the RBC model, the labour market is in equilibrium in each
period, sometimes with positive output-gap, sometimes with
negative output-gap
The business cycles generated by the RBC model are therefore
often called equilibrium business cycles.
Unlike the cycles of the AD-AD model, where there is
downward pressure on nominal price growth as well as on the
real wage level when ut u > 0 , and upward pressure when
ut u < 0.
Another di¤erence is that there is only room for supply shocks
in the RBC model, there are no aggregate demand shocks.
During the last 25 years there has been a synthesis in the form
of so called New-Keynesian models and the RBC model. The
synthesis is called DSGE (dynamic stochastic general
equilibrium) models.
ECON 3410/4410: Lecture 6
RBC and AD-AS model II
The main ingredient of the DSGE is a RBC “core” model for
the real economy, and a new theory of nominal price
adjustment for in‡ation, the so called New Keynesian Phillips
curve.
DSGE has been successful in shaping the thinking of in‡ation
targeting central banks.
The …nancial crisis has started a new controversy. For
example the current “Krugman debate”
Krugman:
www.nytimes.com/2009/09/06/magazine/06Economict.html?_r=1
Cochrane:
http://faculty.chicagobooth.edu/john.cochrane/research
/Papers/krugman_response.htm
ECON 3410/4410: Lecture 6
RBC and AD-AS model III
The Economist features in July.
Lucas’response: http://www.economist.com/business…nance/
displaystory.cfm?story_id=14165405
ECON 3410/4410: Lecture 6