Advanced
Macroeconomics
Chapter 19
EXPLAINING
BUSINESS
CYCLES
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Themes of the chapter
• Explaining business cycles by means of
the AS-AD model.
 The Frisch-Slutzky paradigm: Impulse
and propagation.
 The deterministic versus the stochastic
AS-AD model.
 Supply versus demand shocks.
 Permanent versus temporary shocks.
 The theory of real business cycles.
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Percent
US GDP
US Inflation
5
4
3
2
1
0
-1
-2
-3
-4
-5
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
The cyclical components of real GDP and inflation in the United States
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percent
4
USA
3
2
Germany
1
0
-1
Japan
-2
1996
1997
1998
1999
2000
2001
2002
2003
Growth in real per capita GDP in the USA, Germany and Japan
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The Frisch-Slutzky Paradigm
 The impulses (demand and supply shocks) initiating business cycles
may be unsystematic
 The propagation of the impulses may generate systematic
fluctuations due to the structure of the economy
Basic questions
 Why do movements in economic activity display persistence?
 Why do these movements tend to follow a cyclical pattern?
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Restating the AS-AD model
yt  y  1  gt  g    2  rt  r   vt
rt  it  
e
t 1
(2)
it  r  te1  h  t  *   b  yt  y 
t      yt  y   st
e
t
  t 1
e
t
(1)
(3)
(4)
(5)
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The AS-AD Model in compact form
Inserting (2), (3) and (5) into (1), we get
yt  y     * t   zt ,
vt  1  gt  g 
zt 
1   2b
2h

,
1   2b
(6)
which may be rearranged to give the aggregate demand curve:
AD:
1
t       yt  y  zt 

*
(7)
Substituting (5) into (4), we obtain the short-run aggregate supply curve:
SRAS: t  t 1    yt  y   st
(8)
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
LRAS
SRAS
0
E0
AD
y0
y
y
A short-run macroeconomic equilibrium with cyclical unemployment
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
LRAS
SRAS0
SRAS1 (e = 0)
SRAS2 (e = 1)
SRAS3 (e = )
0
1
2
3
*
AD
y0
y1 y2 y3
y
y
The adjustment to long-run equilibrium
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How long is the Long Run?
Defining
yˆt  yt  y
(output gap)
ˆt   t   *
(inflation gap)
and assuming st = zt = 0, we may rewrite (7) and (8) as
1
AD: ˆt 1     yˆt 1 ,
 
2h

1   2b
SRAS: ˆt 1  ˆt   yˆt 1
(9)
(10)
Inserting (9) into (10), we find
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How long is the Long Run?
1

1  
yˆt 1   yˆt ,
(11)
ˆt 1  ˆt
(12)
The solutions to these linear first-order difference equations
are
yˆt  yˆ 0  ,
t  0,1, 2,.....
ˆt  ˆ0  ,
t  0,1, 2,.....
t
t
(13)
(14)
Note that the long-run equilibrium is stable, since 0 < β < 1.
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The speed of convergence
th  number of periods before half the adjustment to long-run
equilibrium has been completed
1
1
1
th
h
yˆt  yˆ 0   yˆ 0   
 t ln   ln   
2
2
2
th
ln 2
0.693
th  

ln 
ln 
(15)
According to equation (12) in Chapter 17 we have
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Calibrating the model (one time
period = one quarter)
1

1  
h 2

1  b 2
 1  
 Dr
2 



Y0 (1  Dy )  1  Dy 
  0.05
  0.2
Dy  0.8
 Dr

Y0 (1   )
  3.6
From this it follows that   0.958  th  
h  b  0.5
ln 2
 16  4 years
ln 
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
LRAS
SRAS1
s1
SRAS2




E1
SRAS0
E2
E
AD
y1
y2
y
y
Effects of a temporary negative supply shock
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
LRAS
SRAS0



E

SRAS2
E2
E1
AD0 = AD2
AD1
y1
y
y2
y
Effects of a temporary negative demand shock
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y-y
Percent
π - π*
1
0,8
0,6
0,4
0,2
0
-0,2
-0,4
-0,6
-0,8
-1
1
3
5
7
9
11
13
15
17
Year
19
21
23
25
27
29
The adjustment to a temporary negative supply shock
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(s1=1)
y-y
Percent
π - π*
0,2
0
-0,2
-0,4
-0,6
-0,8
-1
1
3
5
7
9
11
13
15
17
Year
19
21
23
25
27
29
The adjustment to a temporary negative demand shock (z1= -1)
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Permanent Shocks
When analyzing permanent shocks, we must account for the fact that such
shocks will change the long-run equilibrium real interest rate (the ’natural’
interest rate). Denoting the initial values of natural output and the natural
interest rate by zero subscripts, we may write our AS-AD model as
yt  y0  vt   2  rt  r0  ,
vt  vt  1  gt  g 
t  t 1    yt  y0   st
(21)
(22)
Consider an initial equilibrium wherevt  st  0 and suppose that st
permanently changes from zero to some s ≠ 0. The new long-run equilibriu
level of output may then be found from (22) by inserting
t  t 1 ,
to get
yt  y ,
st  s
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A permanent supply shock
The effect of a permanent supply shock on natural output
s
y  y0 

(23)
The new equilibrium real interest rate is found from (21) by setting
s
yt  y  y0  ,

rt  r
to get
The effect of a permanent supply shock on the equilibrium real interest
rate
s
r  r0 
 2
(24)
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A permanent demand shock
A permanent demand shock does not affect natural output. Hence the effect on
the equilibrium real interest rate may be found from (21) by setting
yt  y0 , vt  v and rt  r to get
The effect of a permanent demand shock on the equilibrium real interest rate
v
r  r0 
2
(25)
To keep inflation close to its target rate and to avoid large deviations of output
from trend, the central bank must revise the estimates of natural output and of the
equilibrium real interest rate entering the Taylor rule when the economy is hit by
permanent shocks. The adjustment of the economy to the new long-run equilibrium
will depend on how long it takes the central bank to realize the permanency of the
shock. Exercise 19.2 invites you to study these issues further.
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The Stochastic AS-AD model
The deterministic AS-AD model considered above can explain the
persistence in macroeconomic time series, but it cannot explain the
recurrent cyclical fluctuations observed in the real world. To generate such
fluctuations, we now assume:
Stochastic demand and supply shocks
zt 1   zt  xt 1 ,
xt
N (0,  ) ,
2
x
st 1   st  ct 1 ,
ct
N (0,  c2 ) ,
0   1
(27)
xt i.i.d .
0   1
(28)
ct i.i.d .
Our goal is to calibrate a stochastic AS-AD simulation model which can reproduce
the stylized business cycle facts summarized in Table 19.1.
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Percent
5
US GDP
US Inflation
4
3
2
1
0
-1
-2
-3
-4
-5
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
Year
Cyclical components of real GDP and inflation in the USA, 1974-98
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Table 19.1: The stochastic AS-AD model and the stylized business cycle facts
Autocorrelation in output
Standard deviation (%) Correlation between
Autocorrelation in inflation
output and inflation
t-1
t-2
t-3
t-4
t-1
t-2
t-3
t-4
0,08
0,81
0,66
0,47
0,37
0,99
0,96
0,91
0,85
1,90
-1,00
0,92
0,86
0,79
0,73
0,92
0,86
0,79
0,73
1,66
0,30
0,15
0,82
0,68
0,50
0,38
0,47
0,33
0,24
0,32
1,66
0,29
0,10
0,86
0,65
0,41
0,18
0,50
0,29
0,24
0,17
Output
Inflation
1,62
0,52
1,67
AS-AD model with static
expectations and no supply shocks1
AS-AD model with static
expectations and no demand shocks 2
AS-AD model with adaptive
expectations and a combination
of demand and supply shocks3
The U.S economy,
1955:I-2001:IV4
1
4
Φ = 0, σc = 0, σx = 1, δ = 0.75, ω = 0
2
Φ = 0, σc = 0.75, σx = 0, δ = 0, ω = 0
3
Φ = 0.92, σc = 0.25, σx = 1, δ = 0.75, ω = 0.25
The cyclical components of output and inflation have been estimated via detrending of quarterly data using the HP-filter with λ = 1600.
Common parameter values in all AS-AD simulations: γ = 0.05, τ = 0.2, DY = 0.8, η = 3.6, h = b = 0.5
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
Percent
y-y
5
4
3
2
1
0
-1
-2
-3
-4
-5
1
11
21
31
41
51 61
Quarter
71
81
91
Simulation of the stochastic AS-AD model with static expectations and no
supply shocks
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14
12
e
Expected inflation rate for the current quarter (πt )
Actual inflation rate during the previous quarter (πt-1)
10
8
6
4
2
0
-2
-4
1981-III
1984-III
1987-III
1990-III
1993-III
1996-III
1999-III
2002-III
Expected current inflation and lagged actual inflation in the United States
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The Stochastic AS-AD model with
static expectations
Problems
 The model with demand shocks can reproduce the stylized facts
regarding output, but it generates far too much persistence of inflation
 The model with supply shocks is unable to reproduce the stylized
facts of output as well as inflation
● The assumption of static expectations implies greater fluctuations in the
expected inflation rate than what we observe in practice (see Figure 19.12)
To solve these problems we will now allow for simultaneous demand
and supply shocks as well as adaptive expectations.
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Adaptive expectations
revision of expected
inflation rate
last period's inflation
forecast error
 te   te1  (1   ) ( t 1   te1 ) ,
0   1
(29)
For   0 we get static expectations. Eq. (29) may be rewritten as

 te    n 1 (1   ) t  n
(33)
n 1
The AS-AD model with adaptive expectations
AD: yt  y    *  t   zt
(34)
SRAS: t  te    yt  y   st
(35)
Expectations: te  te1  1    t 1
(36)
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The AS-AD model with adaptive
expectations
The model (34) through (36) may be condensed to:
yˆt 1  ayˆt   ( zt 1  zt )   st 1   st
(41)
ˆt 1  aˆt   zt 1   zt   st 1   st
(42)
1  
a
1
1  
1

1
1  
The third row in Table 19.1 shows that this model reproduces the U.S. business
cycle reasonably well, given its simplicity.
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Pe r ce nt
5
4
3
2
1
0
-1
-2
-3
-4

y-y
0
97
93
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
13
9
Quar te r s
5
1
-5
Percent
5
US GDP
US Inflation
4
3
2
1
0
-1
-2
-3
-4
-5
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
Year
The AS-AD model with adaptive expectations (top diagram)
versus the actual U.S. business cycle (bottom diagram)
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The theory of real business cycles
Our AS-AD model of the business cycle emphasizes the role of
expectational errors and sluggish wage and price adjustment, and the
microfoundation for the SRAS curve implies that business fluctuations
are associated with fluctuations in involuntary unemployment. The
model also assigns an important role to demand shocks. A very
different theory is:
Real business cycle theory (basic version)
● The business cycle is mainly driven by fluctuations in the rate of
productivity growth
● The employment fluctuations observed during business cycles reflect
voluntary movements along individual labour supply curves (intertemporal
substitution in labour supply, no involuntary unemployment)
● Economic growth and business cycles can and should be explained within
unified model framework. To explain business cycles, there is no need to
postulate nominal and/or real rigidities.
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A simple RBC model: technology
Production function: Yt  K  At Lt  , 0    1

t
1
Actual productivity: lnAt  gt  st ,
st 1  st  ct 1 ,
0  <1, ct
N  0 ,
(44)
(45)
2
c

Trend productivity: lnAt  gt
(46)
Capital accumulation: Kt  1    Kt 1  St 1
(47)
For simplicity, we will assume that δ = 1.
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A simple RBC model: economic
behaviour
Saving: St  s  Yt , 0  s  1
 wt 
s
Labour supply: Lt    ,   0
 wt 
 Kt 
Yt
Profit maximization: wt 
 1    

Lt
 At Lt 
Trend real wage: wt  1    cAt ,
(49)

 
c k
Labour market clearing: Lt  L
s
t
*
(48)
(50)

(51)
(52)
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A simple RBC model
As shown on pp. 585-86 in the text, the model (44) through (52) may be reduced to


 1  

ˆyt  
 ˆyt 1  
 st ,
 1   1    
 1   1    


1
1 
ˆ  ˆy
L
t
t
Propagation mechanism in the model: A positive productivity shock
(59)
(60)
raises current income which in turn raises saving and capital
accumulation. This leads to a higher capital stock in the next period,
which in turn raises next period’s income and saving, and so on. In this
way a temporary productivity shock generates persistence in output and
employment. Indeed, we see from Table 19.3 that the calibrated version
of the model generates too much persistence compared to the
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persistence observed in the U.S. data.
Standard deviation (%)
Standard deviation of
Autocorrelation
Autocorrelation
output relative to standard
in output
in hours worked
Output
Hours worked
deviation of hours worked
t-1
t-2
t-3
t-1
t-2
t-3
RBC model1
3.42
2.84
0.83
0.75
0.50
0.23
0.75
0.50
0.23
The U.S economy2
3.47
2.88
0.83
0.76
0.38
0.08
0.73
0.29
0.06
α = 0.33, η = 0.83, ω = 0.1, σc =
0.015
1
2
Annual data for the business sector.
Note: The cyclical components of output and employment have been estimated via linear OLS detrending of annual data.
Source: Economic Outlook Database, OECD.
Table 19.3: The RBC model versus the U.S. economy
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Some problems with basic RBC
theory
• The virtue of the basic RBC model is that it is simple
and fully integrates the theory of business cycles with
the theory of economic growth. However, critics
object to the theory by raising the following
questions:
• Is technological progress really so uneven as
postulated in the RBC model?
• Is it really plausible that recessions are periods of
technological regress? (Alternative hypothesis: the
observed fluctuations in productivity reflect
fluctuations in capacity utilization caused by demand
shocks).
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Some problems with basic RBC
theory
• Are the observed fluctuations in employment really a
reflection of intertemporal substitution in labour
supply? To reproduce the observed volatility of
employment relative to the volatility of output, the
wage elasticity of labour supply in our simple RBC
model (ε) has to be set at 4.9 which is much higher
than the elasticity estimated by labour economists.
More generally: Is all recorded unemployment really
voluntary?
• The RBC model predicts that the real wage is
procyclical. This is in line with U.S. data, but it is not
consistent with European data.
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The lasting contribution of real
business cycle theory
In response to these criticisms, real business cycle theorists have
recently tried to make their models more realistic by allowing for various
frictions and rigidities, including (in some cases) nominal rigidities.
At the methodological level RBC theorists have made a lasting
contribution by pointing out that supply shocks may play an important
part in the explanation of business cycles, and by insisting that a
satisfactory theory of the business cycle should consist of a dynamic
stochastic general equilibrium model which is able to reproduce the
most important stylized facts of the business cycle reasonably
well.
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