The dynamic AD-AS model for the closed
economy–Part II
Ragnar Nymoen
Department of Economics, UiO
15 September 2009
ECON 3410/4410: Lecture 5
Lecture notes on:
IAM Ch 18 (but omit 18.3)
IAM Ch 19.1 and 19.2
ECON 3410/4410: Lecture 5
Price setting and labour demand I
Each monopolistic sector faces “its own” demand schedule. 1 o c
for pro…t maximization
marginal revenue = marginal cost
With symbols from IAM ch 18, the 1 o c translates to:
Pi
1
=
mp
(1
|
Wi
, mp =
)BLi
{z
}
1
> 1, ( > 1)
(1)
marginal cost
mp is called the price mark-up coe¢ cient. It depends on the
product demand elasticity .
(1
) is the elasticity of output with respect to employment in
the production function which is
(1
Yi = BLi
)
ECON 3410/4410: Lecture 5
Price setting and labour demand II
Because of monopolistic competition we can have (1
) 1
without violating pro…t maximization.
Dividing by P on both sides of (1), and insertion of (1) back into
the demand equation, and the product function gives IAM equation
(8) which is very important in the rest of the derivations in IAM:
Li =
Y
nB
=
B(1
mp
)
Wi
P
,
=
1+ (
1)
>0
(2)
Here, Y is aggregate output (GDP) and n are the number of
sectors in the economy.
ECON 3410/4410: Lecture 5
Wage setting
Unions in each sector set the nominal wage that maximizes a
weighted sum of expected real wage, and employment in the
sector. In doing so, they take account of price-setting and
labour demand, speci…cally equation (2).
The resulting wage setting equation, (11) in IAM, is
Wi = P e mw b
(3)
where Wi is the nominal wage and b is real compensation in
case of unemployment, P e the expected price level, and the
wage mark-up coe¢ cient mw > 0 depends on the underlying
parameters in the …rms’demand and production function (see
IAM (10)).
We later also use
Wi
Pe w
=
m b,
P
P
ECON 3410/4410: Lecture 5
(4)
The expectations-augmented Phillips curve I
IAM assume that all sectors are symmetric, meaning that price
setting and labour demand in all sectors follow the same
relationships.
Total labour demand is then (see IAM eq (13))
L=n
B(1
) P
p
w
m m b Pe
1=
=
using (4)
n
B(1
mp
)W
P
1=
(5)
In the long-run, unions have correct price expectations:
Pe = P
Insertion of the long-run condition in (5) gives:
L=n
B(1
)
p
w
m m b
1=
ECON 3410/4410: Lecture 5
(6)
The expectations-augmented Phillips curve II
while division of (5) by (6) gives (16) in IAM, namely
L
=
L
P
Pe
1=
(7)
Next introduce u as the rate of unemployment, and N as the
labours force, and also two de…nitional equations:
L = (1
u)N
L = (1
u)N
and
to de…ne the natural rate of unemployment u and the natural rate
of employment L.
We can then manipulate (7) to obtain the expectations augmented
Phillips curve (PCM) relationship
ECON 3410/4410: Lecture 5
The expectations-augmented Phillips curve III
ln(1 u)
| {z }
u
ln P
ln P
1
(1 u)N
P 1=
=
(1 u)N
Pe
ln(1 u) = 1= (ln P
| {z }
ln P e )
u
ln P
e
ln P e
ln P + ln P
1
=
=
=
e
(u
u)
(u
u)
(u
u) , 0
(8)
where ln P 1 refers to the previous period. With t for time period
as usual, we have:
t
=
e
t
(ut
u) , 0
ECON 3410/4410: Lecture 5
(9)
The crucial role of expectations errors in the derivation of
the Phillips curve in IAM I
Note that the assumption about price level expectation errors:
Pte 6= Pt
is essential in this theory of the PCM.
If
Pte = Pt
in all periods we would have:
Pt
=
Wt
(1
mp
)BLt
from price setting, and
Pt
1
= w
Wt
m bt
from wage setting.
ECON 3410/4410: Lecture 5
The crucial role of expectations errors in the derivation of
the Phillips curve in IAM II
This gives:
Lt =
m p m w bt
(1
)B
1=
=
Lt
n
showing that without expectations error with respect to Pt :
1
Employment would (in this model) be equal to natural
employment in each period:
ut = ut
and
2
t
=0
so there would be no in‡ation either.
ECON 3410/4410: Lecture 5
The crucial role of expectations errors in the derivation of
the Phillips curve in IAM III
While it is true that expectation errors, or more generally,
inconsistent expectations between …rms and workers are
important, there are certainly many more factors that cause
in‡ation— demand pressure as such (i.e. beside any e¤ects
that simply mirror expectations errors) is one of them.
We will return to this issue in the last few lectures when we
show that the wage-bargaining model can be given a di¤erent
interpretation, namely as theories about the steady state, and
in that interpretation in‡ation is in‡uenced by a long list of
factors.
ECON 3410/4410: Lecture 5
The wider interpretation of PCM
t
=
e
t
(ut
u) ; 0
has been given a very stringent, but also restrictive,
motivation in IAM, and there are several other ways to
rationalize a similar relationship.
Hence, we will use the expectations augmented Phillips curve
with that wider interpretation in mind.
In particular, note that other derivations suggest that
@ 2 t =@ut2 > 0, so that an increase in ut reduces t less when
the initial level is low compared to an increase that takes
place from a low level of ut . As in Fig 18. 4 in IAM.
This can be represented by writing the short-run Phillips curve
in terms of ln ut .
ECON 3410/4410: Lecture 5
Short-run and long-run Phillips curve I
The short-run Phillips curve is simply
t
=
e
t
(ut
u) , 0
(10)
e
t
may be interpreted as an exogenous variable, but often a
model for expectations (adaptive, static) is speci…ed for the
short-run Phillips curve.
The long-run Phillips curve is de…ned by imposing
t
=
e
t
=
(11)
in (10), which gives
ut = u i¤
t
=
e
t
=
(12)
which we represent as a vertical line through u in the ,u diagram.
ECON 3410/4410: Lecture 5
Short-run and long-run Phillips curve II
A slightly more general short-run Phillips curve is
t
If
='
e
t
(ut
u) , 0 < '
1
(13)
< 1, the long-run Phillips curve is
=
1
'
(u
u) , 0 < ' < 1
which is downward sloping.
It is custom to refer to a model like (10) as a homogenous
Phillips curve, and (13) as a non-homogenous Phillips curve.
ECON 3410/4410: Lecture 5
The accelerationist model I
When the PCM in (10) is combined with an assumption about
backward looking expectations:
e
t
=
t 1
we obtain the homogenous short-run Phillips curve:
t
t 1
=
(ut
u) ,
In this model, the price level accelerates (in‡ation increases) when
ut u < 0 and decelerates when ut u > 0.
In‡ation is constant if and only if ut u.
As we will see at the end of the course, this view is probably too
restrictive
ECON 3410/4410: Lecture 5
Determinants of the natural rate I
Since u is determined by the speci…c macro model used, its
determinants re‡ect the assumptions of that model.
It is model dependent and not natural at all!
Starting from the de…nition
u=1
L
N
and setting N = n we have from the theory in this chapter
that
B(1
) 1=
u=1 L=1
mp mw b
from (6) above.
Higher mark-ups by …rms and workers increase the natural
rate. So institution and policies that reduce these are seen as
bene…cial. “Nordic model” one of them?
ECON 3410/4410: Lecture 5
Determinants of the natural rate II
Higher (total factor) productivity B reduced the natural rate.
But, realistically, B is not constant in the long-run steady
state. It has a constant growth rate.
Hence this model seems to imply a trend-like reduction in u.
To avoid that IAM sets b proportional to B.
ECON 3410/4410: Lecture 5
Supply Shocks
Intuitively, in‡ation may be a¤ected by short-run ‡uctuations
in the mark-ups, mp and mw , and B.
Ch. 18.6 in IAM contains the necessary modi…cations of the
baseline model.
For our purposes it is su¢ cient to represent supply shocks by
adding st to the short-run Phillips curve, as in
t
=
e
t
(ut
u) + st
where we think of st as a random and exogenous variable with
zero mean.
Note that although IAM p 537 uses s, the notation is changed
to st , Iin Vh 19 .
ECON 3410/4410: Lecture 5
The aggregate supply (AS) schedule
As we have seen, we can represent aggregate demand AD in
terms of log GDP deviations from trend: yt y , the
output-gap.
In the model developed in IAM there is a functional
relationship between ut u and yt y .
ut u is function of ln Lt
related to yy y .
ln L which in turn is functionally
Empirically, there is also a close mapping from unemployment
to output gap— this is called Okun’s law.
With reference to theory and Okun’s law, we write the
short-run aggregate supply (AS) schedule as
t
=
e
t
+ (yt
y ) + st ,
> 0:
ECON 3410/4410: Lecture 5
AD-AS model— the equations
yt
y
rt
=
(gt
= it
= r+
it
e
t+j
=
t
=
Mt
Pt
1
g)
e
t+1 ;
e
t+1 +
t+j 1 , for
e
t + (yt
= e m 0 Ytm 1 e
2
(rt
r ) + vt ;
(14)
(15)
h(
t
) + b (yt
y) ;
(16)
j = 0; 1;
(17)
y ) + st :
(18)
m 2 it
:
(19)
(14) and (16), and (17) give the AD schedule for a monetary
policy regime of in‡ation targeting.
(18) and(17) give the short-run AS schedule.
(19) represents the money market equilibrium condition, we
need to change notation from …rst part of Lecture 5, to avoid
con‡ict.
ECON 3410/4410: Lecture 5
AD schedule I
yt
y
=
1
(gt
2 fr
yt
y
=
g)
+
1
e
t+1
(gt
2 fr
(1 +
2 b)(yt
yt
|
y) =
y
+h(
t
) + b (yt
t
) + b (yt
{z
y)
g)
+h(
rt
2h ( t
=
|{z} (
)+
t
1
(gt
y )g + vt
}
) + zt
2h
(1+ 2 b)
zt
=
e
t+1 g
1
(gt g ) + vt
(1 + 2 b)
ECON 3410/4410: Lecture 5
g ) + vt
+ vt
AD schedule II
The AD function is therefore
=
t
1
(yt
y
zt )
with slope
@ t
@yt
=
1
AD
=
(1 +
2 b)
2h
<0
and zt given by:
zt =
1
(gt g ) + vt
(1 + 2 b)
ECON 3410/4410: Lecture 5
(AD)
Taylor rule: Impact on AD slope
π
If y is increased
(from the supply
side) then r
needs to be
reduced.
t
If b = 0 this is
attained by
lower t :
A D ( h h ig h , b lo w )
A D ( h lo w , b h ig h )
yt
ECON 3410/4410: Lecture 5
If b > 0 product
market
equilibrium
requires even
larger reduction
in t
AD-AS, short-run model (i-targeting, static expectations)
=
t
t
mt
pt 1
| {z
pt
=
1
(yt
t 1
y
zt ) ;
+ (yt
it
= r+
}t
= m0 + m1 yt
t
+h(
(AD)
y ) + st ;
t
) + b (yt
m2 it
(SRAS)
y)
(T)
(M-market)
The last equation is the log of (19) and with pt is written as
pt 1 + t .
Endogenous. t , yt ; it and mt .
Exogenous: zt , st , and t 1 (which is pre-determined in
period t).
The model contains a simultaneous block: (AD) and (SRAS),
and a recursive block: (T) and (M-market).
ECON 3410/4410: Lecture 5
AD-AS graphical solution
π
Short-run
equilibrium is at
A
t
LRAS
AD
SRAS
Long-run (steady
state) at B
A
π
*
Is the dynamic
process (from A
to B) stable?
B
Full employment GDP
yt
ECON 3410/4410: Lecture 5
Need
dynamic
analysis to
answer that!
AD-AS long-run model I
De…ned by
z = 0, and
t
=
t 1
=
Correctness of expectations is implied:
t
e
t
=
+
=
t 1
=
e
t+1
=
The LRAS function then entails
y =y
ECON 3410/4410: Lecture 5
AD-AS long-run model II
The static equations that make up the long-run model is therefore:
y
i
= y
(LRAS)
= 0
(AD)
=
)
(i
| {z }
r under
(m
Since y and
model:
+h(
)
(T)
target
p) = m0 + m1 y
m2 i
(M-market)
are determined in the AD-AS block of the long-run
i
(m
+
= i from (T)
+
p) =
m0 + m1 y
m2 i from (M-market)
Therefore the endogenous variables are y , ; i and real money
supply: (m p).
ECON 3410/4410: Lecture 5
The dynamic solution of the model I
Lag the AD equation one period:
=
t 1
1
(yt
y
1
zt
1)
and insert in SRAS:
1
=
t
(yt
1
y
zt
1)
+ (yt
y ) + st ;
Back into AD schedule:
1
(yt
1
y
zt
1)
+ (yt
y ) + st
=
1
(yt
ECON 3410/4410: Lecture 5
y
zt )
The dynamic solution of the model II
Collecting terms:
(yt
y)
1
=
+
+
(yt
1
1
=
=
y) =
1
(yt
1
y ) + (zt
1
1
+1
(yt
1
zt
1)
st
p 564.
y ) + (zt
zt
1)
st(20)
as in eq (19) on p 568.
(20) is a …nal equation, and gives the solution for (yt y )
t = 1; 2; ::::conditional on predetermined y0 y , and z0 and
given sequences of (zt zt 1 ) and st for t = 1; 2; ::3
ECON 3410/4410: Lecture 5
Graphical illustration of the dynamic solution
π
t
LRAS
AD
S R A S0
π
π
SRAS1
A
0
*
B
y0
Full employment GDP
ECON 3410/4410: Lecture 5
yt