• Ch 19 and 20 in IAM
Lecture 8: Aggregate demand and supply in
action, closed economy case.
• Since our primary concern is to develop and apply the methodology of
dynamic analysis we focus on the main properties of the models. This is
not a course in business cycle analysis, so the book’s focus on how well
the AD-AS model fit the business cycle is not emphasized here.
• Ch 20. The main point here is a (simplified) discussion of the book’s
discussion of optimal choice of weights in the Taylor rule.
Ragnar Nymoen
Department of Economics, University of Oslo
October 18, 2007
2
1
The model of aggregate demand and supply (IAM
19.1 and 19.2)
yt = α0 + α1gt − α2rt + vt, α1 > 0, α2 > 0
e
rt = it − πt+1
(1)
Note that mt − pt ≡ πt − pt−1 has been used.
(2)
(5) is the PCM. From the first part of the lectures we remember that this
representation of wage-and price settings strongly restricts how the “nominal
dynamics” (wage and price evolution through time) can be stabilized. Instead
of unemployment, product and labour market “pressure” is represented by the
output gap (yt − ȳ), ȳ represents output at full employment,
e + h(π − π ∗) + b(y − ȳ), h > 0, b > 0
it = r̄ + πt+1
t
t
(3)
πt = πte + γ(yt − ȳ) + st
(5)
mt − πt − pt−1 = m0 − m1it + m2yt, mi > 0, i = 1, 2
Equation (4) represents equilibrium in the money market (4). We prefer to
represent this explicitly since it is otherwise easy to forget the role of money
(once the Taylor rule has been included).
(4)
(1) is the usual linearized IS curve, supplemented with a variable vt that represents arbitrary demand disturbances (“shocks”), possibly due to shifts in
private sector confidence. Compared to IAM ch 19 we collect ȳ, α1ḡ and α2r̄
e is the
in a constant term. (2) is the definition of the real interest rate. πt+1
expected rate of inflation, one period ahead.
Remember that yt, gt, mt and pt are in logs.
(1)-(5) represents a dynamic system of equations.
(3) is the Taylor rule of Ch 17 in IAM.
3
4
The short-run model
In the short run, in period t, the following variables are exogenous: gt, vt, st
e , inflation expectations are also
and pt−1. If we do not specify a model for πt+1
exogenous However, that would leave the model incomplete since it is realistic
that expectations are endogenous also in the short-run. Hence we follow the
book and specify
We next follow the 3-step approach to the analysis of dynamic models:
1. Define the short-run model
e
= πt+j−1, for j = 0, 1
πt+j
2. Define the long-run model (i.e., for a long-run steady state, assuming that
it exists)
(6)
as the 6th equation of the short-run model. The 6 endogenous variables are
yt, it, rt, πt , πte and mt.
We solve the short-run model by using (1), (2), (3) and (6) to obtain the
semi-reduced form equation:
3. The question about dynamic stability of the long-run solution
1
(yt − zt),
α
α2h
with α =
and
1 + α2b
α + vt + α1gt − α2r̄ + (1 − α2b)ȳ
zt = 0
.
1 + α2b
πt = π ∗ −
5
(7)
6
The long-run model
Equation (7), and:
πt = πt−1 + γ(yt − ȳ) + st
(8)
are 2 equations, referred to as the short-run aggregate demand (SRAD) function, and the short-run aggregate supply (SRAS) function. They define the
equilibrium solutions for yt and πt as functions of the exogenous and predetermined variables:gt, vt, st and πt−1.
The solutions for yt, and πt can be substituted back into the Taylor rule to
give the solution for it, and to in the money market equilibrium condition to
give mt.
In each period, the money market is equilibrated with the aid of central bank
market operations, which can change the stock of money instantaneously.
The long-run model is defined for the following definition of a steady-state
situation
πt = πt−1 = π
which implies that expectations are correct in steady state:
πte = πt
The Phillips curve has to hold also in this situation, hence the steady-state
value of yt has to be equal to ȳ,
y = ȳ
(10)
which we express by saying that the long-run aggregate supply function (LRAS)
is vertical, see figure 19.3 in AIR. The corresponding long-run aggregate demand (LRAD) function is
π = π∗ −
7
(9)
8
1
(ȳ − z̄)
α
(11)
where z̄ is given by
α + α1ḡ − α2r̄ + (1 − α2b)ȳ
z̄ = 0
1 + α2b
where ḡ is the constant value of gt and the two shock variables st and vt are
set to zero.
(10) and (11) define the long-run model. There are two endogenous variables
namely π and y. We see immediately that the long-run solution is
The dynamic analysis
SRAD in (7) holds in each period, hence
1
(yt−1 − zt−1),
α
Substitution in the SRAS function gives
πt−1 = π ∗ −
(12)
(13)
1
(yt−1 − zt−1) + γ(yt − ȳ) + st
α
Finally, substitution on the left hand side by the SRAD in (7) gives
if the inflation target is to be attained. But in that case a third variable is
determined in the long-run, for example the equilibrium interest rate r̄. Remember that is a property of this model that the steady-state GDP level is
given from the supply-side. Specifically, think of ȳ as
1
1
(yt − zt) = π ∗ − (yt−1 − zt−1) + γ(yt − ȳ) + st
α
α
which we can write as an ADL equation for yt:
y = ȳ
π = π ∗.
ȳ = −γ0/γ
where γ0 is the constant term in the PCM, which is implicit in the formulation
in equation (5).
πt = π ∗ −
yt =
π∗ −
(14)
1
1
1
α
yt−1 +
(zt − zt−1) +
γ ȳ −
st
1 + γα
1 + γα
1 + γα
1 + γα
(15)
which is similar to (19) in AIR ch 19.
10
9
Dynamic responses to demand shocks.
Since both γ and α are positive (as long as h > 0) we know from the first part
of the lectures that the dynamic model for yt is dynamically stable.
Specifically, since 0 < 1/(1 + γα) < 1, the autoregressive parameter of (15)
satisfies the stability requirement that we know from before (i.e., the IDM part
of the lectures)
We also know that since πt depends on lags of yt−1, and yt depends on πt−1,
we cannot have stability in yt without also stability of inflation.
11
Assume that we are in a steady-state situation initially, in period t − 1, with
st−1 = vt−1 = 0. Then vt = 1 but vt+1 = vt+2... are zero. Hence we have a
temporary demand shock.
Using (15):
∂yt
1 ∂zt
=
>0
∂vt
1 + γα ∂vt
1 ∂yt
1 ∂zt
1
∂yt+1
1
∂zt
=
−
=
<0
(
− 1)
∂vt
1 + γα ∂vt 1 + γα ∂vt
1 + γα 1 + γα
∂vt
1 ∂yt+1
∂yt+2
=
∂vt
1 + γα ∂vt
The first multiplier is positive, but less than one, since interest rates are increased in the period of the shock. The second multiplier is negative, since
the PCM 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.
12
In the case of a permanent demand shock, all the multipliers are positive and
declining. But the long-run multiplier is zero, as can be seen directly from the
long-run model. For multiplier number j + 1:
1
δj−1.
1 + γα
Note that you can “lift” the expressions for the multipliers from IDM, Table
2.1.
The AD-AS model with adaptive expectations (Ch
19.3)
δj =
Inflation responds more smoothly to a temporary demand shock. This due to
two features of the model.
Which properties of the model are dictated by the choice of static inflation
expectations in the basic version of the closed economy AD-AS model?
In order to investigate, consider adaptive expectation instead.
In the short-run model, replace (6) by
1. Demand shocks only affect inflation indirectly, through yt.
e
e ), 0 ≤ φ ≤ 1,
= (1 − φ)(πt−1 − πt−1
πte − πt−1
2. Movements in yt are smoothed by inflation expectations, which weights
heavily in inflation dynamics. See for example figure 19.6 and 19.8 in
IAM.
Take care to study the examples of supply shocks as well.
or, equivalently:
e + (1 − φ)π
πte = φπt−1
t−1.
14
The model with a downward sloping long-run Phillips curve
With adaptive expectations the SRAS function
replaces (8).
(17)
Note first the SRAD is unaffected since in the Taylor rule has been specified
e cancels out when we substitute to derive the SRAD
in such at way that πt+1
function.
13
e + (1 − φ)π
πt = φπt−1
t−1 + γ(yt − ȳ) + st
(16)
(18)
The PCM above is a special case of
πt = γ eπte + γ(yt − ȳ) + st, 0 < γ e ≤ 1.
Short-run model.
Since the SRAD function is unchanged, and SRAS is given by (18) we conclude
that the short-run model is unaffected by changing the model of expectations.
The impact multipliers are therefore unaffected.
Long-run model
It is also the same as with static expectations. This generic, to models with
expectations: the long-run model is always unaffected by changes in the specification of how expectations are formed.
The dynamic analysis
With a non-homogeneous PCM, the case of γ e < 1, the AD-AS framework
has the same short-run and dynamic properties as above. The long-run model
becomes
1
π = π ∗ − (ȳ − z̄)
α
π = γ eπ + γ(y − ȳ), 0 < γ e < 1
since the LRAS function is no longer vertical but upward sloping.
A main criticism of model with a downward sloping long-run Phillips curve is
that it implies that there is a long-run trade off between inflation and output
(or unemployment), which is now regarded as naive policy optimism.
e
This is affected, since πt−1
enters into the model, not only πt−1. Intuitively
however, dynamic stability is not endangered.
However, this conclusion is avoided in the AIR specification of the AD-AS
model, since the Taylor rule for monetary policy ties down y = ȳ in any case.
15
16
Stabilization policy in the closed economy model (Ch
Real business cycle theory (ch 19.4)
20 in IDM)
The RBC model is very relevant as a competitor to the AD-AS model in the
explanation of business cycles.
It is less relevant for discussion of stabilization policy since the RBC model sees
macroeconomic fluctuations as equilibrium phenomena.
For example, unemployment, and underemployment, are not meaningful concepts.
In the model of Ch 19, with all its simplifications, nevertheless captures the
gist of the standard model for policy analysis in a closed economy.
Since the model economy is dynamically stable, and full employment GDP is
independent of both monetary and fiscal policy, the remaining rationale for
doing macroeconomic policy is to reduce the welfare losses which are due to
the temporary disturbances in demand and supply, and their propagation.
Since our focus is not business-cycle analysis as such, we treat ch 19.4 as
cursory. It gives good training in using the concepts of dynamics, and in
deriving an ADL equation for yt in the way we have seen above.
Welfare losses are assumed to be linked to variations in demand and in inflation.
17
18
In this course we need to bypass the calculation of standard deviations. However, a discussion along the same lines can utilize the dynamic multipliers,
which of course depends on the shocks, but also on the parameters of the
Taylor-rule. Graphical analysis of the AD-AS model can also be used.
Or, it can neutralize the response of yt to any given increase in ztby choosing
a high h. This follows since α
IAM, ch 20.1, formalizes this by assuming that the government seeks to minimize the “sum of standard deviations” in output and inflation, see equation
(1) on page 599.
α=
α2h
1 + α2b
The main question to answer is how the parameters b and h in the Taylor rule
can be chosen to avoid unwanted variability.
is increasing in h.
Policy response to demand shocks
If the government instead has a preference for inflation stabilization, we can
use (8) and (19) to obtain
y
We derived the multipliers for a demand shock above. Denote them by δo,v ,
y
δ1,v , etc. The first multiplier:
π = γδ y and
δo,v
o,v
π = δ π + γδ y .
δ1,v
o,v
o,v
1 ∂zt
∂zt
1
, with
=
(19)
1 + γα ∂vt
∂vt
1 + α2b
If the government has a strong preference for output stabilization, then it
chooses b to be sufficiently high so that the demand shock triggered by the
increase in vt is very small.
Hence, a policy which stabilizes output also reduces the variability of inflation.
There is no trade-off when the source of variability is demand shocks, see Table
20.2 and the associated text.
19
20
y =
δo,v
Policy response to supply shocks
In theory a balanced choice of b and h values are chosen by minimize the welfare
loss function.
Using (8) and (19) again. we obtain:
α
1 + γα
1
y
δ1,s =
δy
1 + γα o,s
y =−
δo,s
The Taylor principle
In most constellations we have h > 0 as a “good” choice of weight on the
inflation term in the Taylor-rule.
and
π = 1 + γδ y
δo,s
o,s
Using the expression for α = α2h/(1 + α2b), we see that the absolute value
of δo,s can be reduced by choosing h as low¯ as ¯“possible”, while maintaining
¯ y ¯
dynamic stability. b > 0 also helps reducing ¯δo,s¯.
In the model specification with static inflation expectations this implies that
the nominal interest rate increases with more than one percentage point if πt
increases by one percentage point.
y
However, if the preference is for inflation stability, it is preferred that δo,s is “as
negative as possible”, and this suggest h > 0 and b < 0.
Therefore the real interest rate increases in the period of the “inflation shock”.
Hence, in the face of supply shocks, there is a trade off between inflation
stabilization and output stabilization.
21
Rules vs discretion.
In the AD-AS model in Ch 19 and 20, there is a Taylor-rule for monetary policy.
A version with “full discretion” would be with it exogenous (no Taylor rule).
The interest rate can then be set on a period to period basis, or with respect
to different goals (or targets) from period to period.
In modern macro economic policy rules have become popular.
Taylor rule, “fiscal policy rules” (Norway for example).
There are good reasons for this: pressure groups (Blair delegating interest rates
to BOE); a stable framework for policy; transparency, tie oneself to the mast
(Ulysses and the sirens), and so on.
However, since what is a good rule depends on circumstances, there is a premium on some discretion if it can be attained without loss of credibility.
23
22