A dynamic AS-AD Model
(Lecture Notes, Thomas Steger, University of Leipzig, winter term 10/11)
This file describes a dynamic AS-AD model. The model can be employed to assess the dynamic consequences of macroeconomic shocks (fiscal policy shocks, monetary policy shocks, and supply shocks). This AS-AD model can be viewed as a
simple business cycle model. It can explain recurrent business cycle fluctions in response to ongoing shocks. (Extension:
reformulate the model by using a Taylor rule to describe monetary policy.)
Reference: Shone, R., Economic Dynamics, CUP, 2002, Chapter 11.
The model
ü Aggregate demand (IS-LM model)
The goods market (IS-curve)
C  C0  b 1   Y
i   
I  I0  h
(1)
e
(2)
real interest rate
Y  CIG
(3)
Notation: C: real consumption, Y : real income (output), I: real investment, i: nominal interest rate, pe : expected inflation
rate,...
The money market (LM-curve)
  Md  ekY edi 
md  k Y  d i
ms  m  p
 Ms 
(4)
M
(5)
P
ms  md
(6)
Notation: md : (natural) logarithm of real money demand, ms : logarithm of real money supply, p: logarithm of the price
level,...
Solving for Y gives (the solution for i is suppressed)
Y
Defining a0 :
C0  I0  G 
h
d
m  p  h e
1  b 1   
C0 I0 G
1b 1
hk
, a1 :=
(7)
hk
d
hd
1-b1-t+
Y  a0  a1 m  p  a2 e ,
d
hk
, and a2 :
d
1b 1
a0 , a1 , a2  0
This equation represents an aggregate demand (AD) curve.
h
hk
this equation can be written as
d
(8)
2
Dynamic_AS_AD_Model.nb
ü Aggregate supply
Aggregate supply is described by
   Y  Yn   e ,
0
(9)
Notation: p: inflation rate, pe : expected inflation rate, Y : real output, Yn : real potential output.
Notice that this AS-curve results from an augmented Phillips curve, p = pe - aU - Un , together with Okun’s law,
U - Un = -bY - Yn , where both a, b > 0.
Alternatively, the following version of the Lucas supply curve Yt = Yn + hPt - Pte  is also compatible with this AS-curve.
Subtratcing and adding h Pt-1 on the RHS gives Yt = Yn + hPt - Pt-1 - Pte - Pt-1 . If Pt denotes the logarithm of the price
level, one can write Yt = Yn + hpt - pet .
ü Expected inflation
We assume that expectations are formed according to an adaptive expectations scheme
e
     e ,
0
(10)
°
e
When the actual rate of inflation exceeds the expected rate (p - pe > 0), expectations are revised upwards (p > 0), and vice
versa.
ü Summarizing: the complete model
Y  a0  a1 m  p  a2 e
   Y  Yn   
e
     e 
(11)
e
(12)
(13)
Model analysis
ü Preliminaries: eliminating one equation
We first take the time derivative of the AD-equation to yield


e
Y  a1 m    a2 
(14)
Next rewrite the AS-equation as p - pe = a Y - Yn  and plug the right-hand side (RHS) into the expected inflation schedule
°
pe = b p - pe , which gives
e
(15)
    Y  Yn 
Finally, we plug the RHS of the AS-curve, equ. (12), and the RHS of equ. (15) into equ. (14).


Y  a1 m   Y  Yn   e   a2   Y  Yn 


Y  a1

Y  a1
(16)
e


m  a1  Y  Yn   a1 e  a2   Y  Yn 

m   a1  a2  Y  Yn   a1 e
ü The reduced-form model: two differential equations
The reduced-form model is described by (plus two boundary conditions)
(17)
(18)
Dynamic_AS_AD_Model.nb
e
    Y  Yn 


Y  a1 m   a1  a2  Y  Yn   a1 e
°
°
e è
This system can be solved for the steady state ( ,Y ), defined by pe = 0 and Y = 0. The steady state reads as follows
 
e 
  m implying :   m

Y  Yn
Numerical model evaluation
ü The steady state
In[2]:=
In[3]:=
In[4]:=
paramInitial  C0  10, I0  5, G  5, b  0.8,   0.3,
h  0.1, k  0.05, d  0.05,   0.1,   1, mdot  0.01, Yn  1;
paramFinal  C0  10, I0  5, G  5, b  0.8,   0.3,
h  0.1, k  0.05, d  0.05,   0.1,   1, mdot  0.01, Yn  1;
a0, a1, a2  
C0  I0  G
1  b 1   
hk
d
,
hd
1  b 1   
h
,
hk
d
1  b 1   
hk
d
;
AS  GraphicsLineYn . paramInitial, 0, Yn . paramInitial, 0.15,
AxesLabel  Y, e , PlotRange  0, 0.06;
 a1  a2 
AD  Plotmdot 
Y  Yn . paramInitial, Y, 0.5, 1.5,
a1
AxesLabel  Y, e , PlotRange  0, 0.06;
p1  ShowAS, AD, Axes  True, AspectRatio  2  3;
è
e 
The steady state (long run equilibrium) is characterized by   m and Y = Yn !
ü Dynamic responses
At first, we express the differential equations in Mathematica notation
In[7]:=
In[9]:=
In[10]:=
In[11]:=
(19)
(20)
(21)
(22)
°
°
To illustrate, below we plot the two curves pe = 0 and Y = 0 in (pe ,Y )-plane.
In[1]:=
3
dee  e't    Yt  Yn;
deY  Y 't  a1 mdot   a1  a2  Yt  Yn  a1 et;
esol, Ysol  mdot, Yn . paramInitial;
ns  NDSolvedee, deY, e0  1.1  esol, Y0  1.1  Ysol . paramFinal,
et, Yt, t, 0, 100;
ent_ : Evaluatens1, 1, 2
Ynnt_ : Evaluatens1, 2, 2
In[13]:=
t_ :  Ynnt  Yn  ent .
paramFinal this defines the actual inflation rate
In[14]:=
Needs"PlotLegends`"
4
Dynamic_AS_AD_Model.nb
In[22]:=
Plotent, t, mdot . paramFinal, t, 0, 30,
AxesLabel  t, "e blue,red", PlotRange  0, 0.025, AxesOrigin  0, 0
PlotYnnt, 1, 1, t, 0, 30, AxesLabel  t, Y,
PlotRange  0.93, 1.1, AxesOrigin  0, 0.93
pe blue,pred
0.025
0.020
0.015
Out[22]=
0.010
0.005
0
5
10
15
20
25
30
25
30
t
Y
1.10
1.05
Out[23]=
1.00
0.95
0
In[17]:=
5
10
15
20
t
AS  GraphicsLineYn . paramFinal, 0, Yn . paramFinal, 0.15,
AxesLabel  Y, e , PlotRange  0, 0.03;
 a1  a2 
Y  Yn . paramFinal, Y, 0.5, 1.5,
AD  Plotmdot 
a1
AxesLabel  Y, e , PlotRange  0, 0.03;
p2  ShowAS, AD, Axes  True, AspectRatio  2  3;
Dynamic_AS_AD_Model.nb
In[20]:=
5
tra  ParametricPlotYnnt, ent, t, 0, 100, AxesLabel  Y, e ,
PlotRange  0.9, 1.13, 0, 0.03, AspectRatio  2  3;
Show
tra,
p1,
p2
pe
0.030
0.025
0.020
Out[21]= 0.015
0.010
0.005
0.000
0.90
0.95
1.00
1.05
1.10
Y
ü How to model shocks
°
(1) Monetary policy shock: change m (Ø set of parameters)
(2) Fiscal policy shock (permanent increase in G): change initial conditions (Y -dimension: Y = a2 pe + a1 m - p + a0 ; pe dimension: pe = const.) (Ø initial conditions; increase G in the set of parameters)
(2) Supply side shock: change Yn (Ø set of parameters)
Results
ü Monetary expansion (code)
6
Dynamic_AS_AD_Model.nb
ü Graphical output
In[257]:=
Showtra, p1, p2
timepath
timepathY
pe
0.04
0.03
Out[257]= 0.02
0.01
0.00
0.90
0.95
pe blue,pred
0.025
1.00
1.05
1.10
1.15
Y
0.020
Out[258]=
0.015
0.010
0
5
10
15
20
25
30
25
30
t
Y
1.05
1.04
1.03
1.02
Out[259]=
1.01
1.00
0.99
0.98
0
5
10
15
20
t
Dynamic_AS_AD_Model.nb
ü Fiscal expansion (code)
ü Graphical output
In[432]:=
Showtra, p1, p2
timepath
timepathY
pe
0.030
0.025
0.020
Out[432]= 0.015
0.010
0.005
0.000
0.90
0.95
pe blue,pred
0.030
1.00
1.05
1.10
1.15
Y
0.025
0.020
Out[433]=
0.015
0.010
0.005
0
5
10
15
20
25
30
25
30
t
Y
1.10
1.05
Out[434]=
1.00
0.95
0
5
10
15
20
t
7
8
Dynamic_AS_AD_Model.nb
ü Negative supply shock (code)
ü Graphical output
In[409]:=
Showtra, p1, p2
timepath
timepathY
pe
0.025
0.020
0.015
Out[409]=
0.010
0.005
0.000
0.90
0.95
pe blue,pred
0.018
1.00
1.05
1.10
Y
0.016
0.014
Out[410]=
0.012
0.010
0.008
0.006
0
5
10
15
20
25
30
25
30
t
Y
1.00
0.99
0.98
0.97
Out[411]= 0.96
0.95
0.94
0.93
0
5
10
15
20
t
Dynamic_AS_AD_Model.nb
9
Phase diagram and economic intuition
°
The mechanics of the model can be understood by discussing the phase diagramm, which shows the two curves pe = 0 and
°
Y = 0 in (pe ,Y )-plane. In addition, the arrows indicate the direction the economy moves to given a position in (e ,Y )-plane.
pe
0.06
°e
p =0
0.05
II
0.04
I
0.03
0.02
III
°
Y =0
IV
0.01
0.7
0.8
0.9
1.0
1.1
1.2
1.3
Y
°
°
Consider the following Y -equation and the pe -equation

  YYn 
Y  a1 m 

change in AD due
to changes in MP
e

a2
  YYn 
e

change in AD due to changes
in real interest rate
e
    Y  Yn 
Comments
(23)
(24)
°
°
°
°
°
1. Above (to the right of) the Y = 0 locus, we have Y < 0, i.e. a1 m - p + a2 pe < 0. To illustrate, start on the Y = 0
locus and increase Y , holding pe fix. This decreases the RHS of (23), via p, by more than it increases the RHS of
°
°
(23), via pe , provided that a1 a > b a a2 ! Consequently, Y falls. Alternatively, increasing pe (starting on the Y = 0
locus and holding Y const.) increases p and thus reduces AD.
°
°
°
2. To the right of the pe = 0 locus, Y - Yn > 0 implying that pe > 0; recall: pe = a b Y - Yn .
An alternative model structure