The dynamic AD-AS model for the closed
economy–Part I
Ragnar Nymoen
Department of Economics, UiO
8 September 2009
ECON 3410/4410: Lecture 5
Notes on reading
We now lecture from IAM, and use the notation in that book.
Ch 14 is a good background chapter, and it motivates the
“short-run” analysis. The more technical parts about business
cycles can be skipped.
Ch 15, on the “consumption function”, and Ch 16 on
investments, also contain valuable background material. Note
that in both chapters, there are many arguments for dynamic
relationships, but in order to keep the macro model tractable
we will abstract from virtually all of them!
We start with Ch 17 here.
So important to keep in mind that more realistic models will
have more complex dynamics than we encounter in the macro
models that we formulate here.
ECON 3410/4410: Lecture 5
Aggregate demand relationships
General budget equation:
Yt = C t + I t + Gt ,
(1)
where Yt , is real GDP, Ct real private consumption, and I
private real investment, and Gt government expenditure and
investments.
t denotes time period.
Behavioural equations:
Ct
It
Gt
= C (Yt
Tt ; rt ; "t );
(2)
= I (Yt ; rt ; "t )
(3)
= Tt
(4)
T is real taxes, r is a real interest rate, and " represents
“business con…dence”.
t
Partial derivatives are denoted CY = @(Y@C
etc. See IDM p
t Tt )
499 for details.
ECON 3410/4410: Lecture 5
Product market equilibrium
We assume that in the short-run GDP is determined by
aggregate demand ad given by (1)-(4). The goods market
equilibrium condition is:
Yt = D(Yt ; Gt ,rt ; "t ) + Gt
{z
}
|
private demand
where the D(:) function has partial derivatives:
DY
= CY + IY ; 0 < DY < 1.
DG
=
Dr
= Cr + I r < 0
CY < 0
D " = C" + I " > 0
(5) de…nes Yt as a function of Gt , rt and "t .
ECON 3410/4410: Lecture 5
(5)
Aggregate demand (AD) function
We write this Aggregate Demand (AD) function as
e t ,rt ; "t )
Yt = D(G
(6)
with derivatives given by implicit derivation of (5):
eG
D
=
1
(1
1 D
| {z Y}
CY ) = m(1
~
CY ) ;
m
~
e r = mD
D
~ r
e
D" = mD"
~
We next assume that a steady-state equilibrium exists for the
system of which (6) is a part.
What does this assumption amount to?
ECON 3410/4410: Lecture 5
AD as deviation from steady-state I
Y , C etc. denote steady-state values.
Equation (6) must also hold in a steady-state, meaning that
e G ; r ; ")
Y = D(
(7)
is an equation in the long-run version of the model we are setting
up.
In IAM they prefer to work with the short-run model in terms of
deviations from steady-state.
Use the general approximation that
Yt
Y
Y
e
ElG D
e
= ElG D
Gt
G
G
Gt
G
G
e
+ Elr D
er r
+D
Y
rt
r
r
rt
r
r
e
+ El" D
e
+ El" D
ECON 3410/4410: Lecture 5
"t
"
"
"t
"
"
AD as deviation from steady-state II
e and El" D,
e evaluated at steady-state values
Assume that ElG D
Y , G and ", are constant parameters across the business-cycle,
er r :
e r 1 is more stable than D
but that the semi-elasticity D
Y
Y
Finally, to express the variables in logarithms, we use that
Yt
Y
Y
ln(Yt )
ln(Y )
etc., and using yt = ln(Yt ) etc., we …nally have
yt
y
e (g
ElG D
| {z } t
1
e r r (rt
g) + D
| {zY}
2
e (ln "t
r ) + El" D
|
{z
which is equation (11) on page 501 in IAM.
ECON 3410/4410: Lecture 5
vt
ln ")
}
(8)
Money market equilibrium— money supply targeting I
IAM write money demand as
Mt
= kYt e
Pt
it
,
> 0;
> 0.
The supply of real money is written as:
Mt
(1 +
=
Pt
(1 +
t )Mt 1
t )Pt 1
where t is the nominal growth rate of money, and denotes
in‡ation (a rate in this case).
We regard
t
and Yt as exogenous on the money market.
ECON 3410/4410: Lecture 5
Money market equilibrium— money supply targeting II
If the central bank targets money supply, then t is also
exogenous, money supply is exogenous in period t, and by
equation supply and demand we get
it =
(
t
t)
+
ln Yt +
ln k
1
(ln Mt
1
ln Pt
If the market was initially in equilibrium, in period t
obtain
it =
(
t
t)
+
ln Yt +
it = i +
(
t
ln k
t)
+
1
ln k + ln Y
(yt
y)
1)
1, we
i
(9)
which is the same equation as (20) on page 505 in IAM, since
from (20) we obtain (9) by setting r = i
as stated at the
top of p 505.
ECON 3410/4410: Lecture 5
Money market equilibrium— money supply targeting III
ln M
t
− ln P
t
m oney supply
Increased m oney growth, or reduced inflation
m oney dem and
Increased
yt
it
ECON 3410/4410: Lecture 5
“Quantitative easing”
Money supply targeting represents a monetary policy regime.
The operative (intermediate) target of monetary policy is the
growth rate of money supply (can set t = in (9)) which
can be controlled by market operations, which is the policy
instrument.
The problem with this regime, as we shall see later, is that the
control of money supply may be illusive in small open
economies.
Under the current credit crises, a related policy has appeared
in the form of “quantitative easing”, which corresponds to
increased money supply in our model: Quantitative easing
seeks to
Reduce the di¤erence between the central banks lending rate
and the market interest rate (it increased when the interbank
market collapsed).
Encourage banks to lend money even when interest rates very
low (“zero”)
ECON 3410/4410: Lecture 5
The Taylor-rule I
If the target for monetary policy is the stabilization of in‡ation
and output— the interest rate it becomes the instrument of
monetary policy.
The Taylor-rule is a function that describes how the central
bank responds to changes in t and yt :
it = i + (h + 1) (
where
t
) + b (yt
y ) , h > 0, b > 0
(10)
is the in‡ation target.
This is very similar to (9), the di¤erence is the explicit
in‡ation target,
and that the parameters h and b are determied by political
preferences, not the structure of money demand.
ECON 3410/4410: Lecture 5
The Taylor-rule II
it =
(i
)
| {z }
r under
+
t
+h(
t
) + b (yt
y ) , h > 0, b > 0
target
The similarity between (9) and (10), only the de…nition of the
constant is di¤erent, is convenient when we later want to
compare how the model economy responds to shocks (under
m-targeting and under -targeting.
h > 0 implies that the real interest rate is increased when
is increased. This is called the Taylor principle
ECON 3410/4410: Lecture 5
t
The expected real interest rate
IAM makes the important precision that the r variable that
a¤ects private real demand is the ex-ante or expected real
interest rate, which is
e
t+1 ,
rt = it
where et+1 denotes the expected rate of in‡ation one period
ahead.
Expectations are made at the end of period t.
The Taylor-rule is modi…ed accordingly
e
t+1
it
= r+
rt
= r +h(
t
+h(
t
) + b (yt
) + b (yt
y)
y)
(11)
(12)
see eq (30) and (31) on page 514.
Because of et+1 , the model of the demand side is going to be
dynamic!
ECON 3410/4410: Lecture 5