Lecture 8: Aggregate demand and supply
dynamics, closed economy case.
Ragnar Nymoen
Department of Economics, University of Oslo
October 20, 2008
1
• Ch 17, 19 and 20 in IAM
• 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.
• The focus is instead on the representation of the transmission mechanism
of monetary policy: How changes in the monetary policy instruments affects the real economy. The transmission mechanisms resides both on the
demand side and on the supply side of the economy.
• Ch 20: The main point here is a (simplified) discussion of the book’s
discussion of optimal choice of weights in the Taylor rule.
2
Aggregate demand theory: Private consumption and
investment
In the following we will use a simple representation (as in IAM):
Ct = C(Yt − Tt, rt), CY > 0, Cr < 0,
(1)
where
C = Private consumption in real terms (the price level of the base year).
Y = GDP in real terms,
T = The tax level, in real terms.
r = The real interest rate:
e
rt = it − πt+1
i = Nominal interest rate
π e = The expected rate of inflation
3
(2)
In IDM we had a consumption function with fewer explanatory variables, essentially only Yt − Tt.
We could have added rt, for example, but left it out for simplicity.
On the other hand, IDM had lagged Ct of course!
The specification here (and in IAM) thus assumes that adjustment of Ct to
changes in Yt − Tt for example is all done within the period of analysis.
The important extension compared to IDM is the real interest rate rt which
has a negative derivative to capture intertemporal substitution.
Note also the definition of the real exchange rate: it is in terms of inflation
expectations.
4
It = I(Yt, rt),
IY > 0, Ir < 0.
(3)
Investment is growing when output is increased because there are strong incentives to increase capacity in such a growth situation (with increased capacity,
the expected dividends to shareholders become higher).
A higher real interest rate reduces the present value of expected dividend increases, and reduces the incentive to invest.
Increased rt also makes other types of investment more attractive compared to
investment in physical capital (bond, bank deposits).
5
The aggregate demand function and the IS curve
We define the aggregate private demand function Dt as
Dt ≡ C(Yt − Tt, rt) + I(Yt, rt) = D(Yt, Tt, rt)
with derivatives:
t
0 < DY ≡ ∂D
∂Yt ≡ CY + IY < 1
t ≡ C +I < 0
Dr ≡ ∂D
r
r
∂r
t
(4)
t
DT ≡ ∂D
∂Tt ≡ −CY < 0
If we let Gt denote public sector expenditure (in real terms) the equilibrium
condition on the product market becomes
Yt = D(Yt, Tt, rt) + Gt
(5)
which defines Yt the IS-curve: the downward sloping curve between GDP and
interest rate, known from the Keynesian macro model of earlier courses.
6
(Graph from IAM with negative relationship between interest rate and aggregate demand)
7
Balanced government budget
We know from before that (in this model) a balanced change in the budget has
an expansionary effect on GDP. For simplicity, we will impose this in the form
of:
T =G
so that the IS curve we will use in the following is
Yt = D(Yt, Gt, rt) + Gt
(6)
∂Yt
≡ −CY + 1 > 0
∂Gt
(7)
meaning that
from the demand side.
8
Steady state and linearization
Our model of aggregate demand is now:
Yt = D(Yt, Gt, rt) + Gt
(8)
and
e .
rt = it − πt+1
Which, despite all the simplifications, is a dynamic model of the demand side,
because of the expectations variable (remember expectations as a “source of
dynamics”).
In the case where all the arguments in the demand function are at their long-run
(steady state) values, the long-run aggregate demand function becomes:
Ȳ = D(Ȳ , Ḡ, r̄) + Ḡ
the IAM textbook shows how (8) can be log-linearized around the steady state
to give
yt − ȳ = α1(gt − ḡ) − α2(rt − r̄) + vt, α1 > 0, α2 > 0
(9)
where yt = ln(Yt), etc., and we introduce vt as random demand-shock variable.
9
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
e + h(π − π ∗) + b(y − ȳ), h > 0, b > 0
it = r̄ + πt+1
t
t
mt − πt − pt−1 = m0 − m1it + m2yt, mi > 0, i = 1, 2
πt = πte + γ(yt − ȳ) + st
(10)
(11)
(12)
(13)
(14)
(10) is 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 in
e is the
a constant term. (11) is the definition of the real interest rate. πt+1
expected rate of inflation, one period ahead.
(12) is the Taylor rule of Ch 17 in IAM.
10
The ‘Taylor rule’ equation
e + h(π − π ∗) + b(y − ȳ), h > 0, b > 0
it = r̄ + πt+1
t
t
represents how a central bank that has the nominal interest rate as its policy
instrument, puts some weight on inflation stabilization, π ∗ denotes the inflation
target, and some on GDP stabilization.
The coefficient h and b reflect policy priorities and preferences.
Note that the central bank puts separate weight on inflation expectations, so
e
goes up, the interest rate is increased (even before) actual inflation is
if πt+1
affected. In this sense, the Taylor-rule captures forward lookingness of interest
rate setting.
11
Equation (13) of the model represents equilibrium in the money market (13).
We prefer to represent this explicitly since it is otherwise easy to forget the role
of money (once the Taylor rule has been included).
Note that mt − pt ≡ mt − πt − pt−1 has been used.
(14) is the Phillips curve model and represents the supply side. 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,
Remember that yt, gt, mt and pt are in logs.
(10)-(14) represents a dynamic system of equations.
12
We next follow the 3-step approach to the analysis of dynamic models:
1. Define the short-run model
2. Define the long-run model (i.e., for a long-run steady state, assuming that
it exists)
3. The question about dynamic stability of the long-run solution
13
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
e
= πt+j−1, for j = 0, 1
πt+j
(15)
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 (10), (11), (12) and (15) to obtain the
semi-reduced form equation:
1
(yt − zt),
α
α2h
with α =
and
1 + α2b
α + vt + α1gt − α2r̄ + α2bȳ
.
zt = 0
1 + α2b
πt = π ∗ −
14
(16)
Equation (16), and:
πt = πt−1 + γ(yt − ȳ) + st
(17)
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.
15
The long-run model
The long-run model is defined for the following definition of a steady-state
situation
πt = πt−1 = π
(18)
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
(19)
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
1
∗
π = π − (y − z̄)
α
16
(20)
where z̄ is given by
α0 + α1ḡ − α2r̄ + α2bȳ
1 + α2b
where ḡ is the constant value of gt and the two shock variables st and vt are
set to zero.
z̄ =
(19) and (20) define the long-run model. There are two endogenous variables
namely π and y. We see immediately that the long-run solution is
y = ȳ
π = π∗
(21)
(22)
if the inflation target is to be attained. 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
ȳ = −γ0/γ
where γ0 is the constant term in the PCM, which is implicit in the formulation
in equation (14).
17
The dynamic analysis
SRAD in (16) holds in each period, hence
1
∗
πt−1 = π − (yt−1 − zt−1),
α
Substitution in the SRAS function gives
1
(yt−1 − zt−1) + γ(yt − ȳ) + st
α
Finally, substitution on the left hand side by the SRAD in (16) gives
πt = π ∗ −
1
1
∗
∗
π − (yt − zt) = π − (yt−1 − zt−1) + γ(yt − ȳ) + st
α
α
which we can write as an ADL equation for yt:
1
1
α
α
yt =
yt−1 +
(zt − zt−1) +
γ ȳ −
st
1 + γα
1 + γα
1 + γα
1 + γα
which is similar to (19) in AIR ch 19.
18
(23)
(24)
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 (24)
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, we
cannot have stability in yt without also stability of inflation.
19
Dynamic responses to demand shocks.
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 (24):
∂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.
20
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 =
δj−1.
1 + γα
Note that you can “lift” the expressions for the multipliers from IDM, Table
2.1.
Inflation responds more smoothly to a temporary demand shock. This due to
two features of the model.
1. Demand shocks only affect inflation indirectly, through yt.
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.
21
The AD-AS model with adaptive expectations (Ch
19.3)
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 (15) by
e
e ), 0 ≤ φ ≤ 1,
= (1 − φ)(πt−1 − πt−1
πte − πt−1
(25)
or, equivalently:
e + (1 − φ)π
πte = φπt−1
t−1.
(26)
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.
22
With adaptive expectations the SRAS function
replaces (17).
e + (1 − φ)π
πt = φπt−1
t−1 + γ(yt − ȳ) + st
(27)
Short-run model.
Since the SRAD function is unchanged, and SRAS is given by (27) 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
e
enters into the model, not only πt−1. Intuitively
This is affected, since πt−1
however, dynamic stability is not endangered.
23
The model with a downward sloping long-run Phillips curve
The PCM above is a special case of
πt = γ eπte + γ(yt − ȳ) + st, 0 < γ e ≤ 1.
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
e
π = γ π + 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.
However, this conclusion is avoided in the IAM specification of the AD-AS
model, since the Taylor rule for monetary policy ties down y = ȳ in any case.
24
Stabilization policy in the closed economy model (Ch
20 in IDM)
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.
Welfare losses are assumed to be linked to variations in demand and in inflation.
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.
25
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.
The main question to answer is how the parameters b and h in the Taylor rule
can be chosen to avoid unwanted variability.
Policy response to demand shocks
y
We derived the multipliers for a demand shock above. Denote them by δo,v ,
y
δ1,v , etc. The first multiplier:
1 ∂zt
∂zt
1
, with
=
(28)
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.
y =
δo,v
26
Or, it can neutralize the response of yt to any given increase in ztby choosing
a high h. This follows since α
α=
α2h
1 + α2b
is increasing in h.
If the government instead has a preference for inflation stabilization, we can
use (17) and (28) to obtain
π = γδ y and
δo,v
o,v
π = δ π + γδ y .
δ1,v
o,v
o,v
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.
27
Policy response to supply shocks
Using (17) and (28) again. we obtain:
α
1 + γα
1
y
y
δ1,s =
δo,s
1 + γα
y =−
δo,s
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¯.
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.
Hence, in the face of supply shocks, there is a trade off between inflation
stabilization and output stabilization.
28
In theory a balanced choice of b and h values are chosen by minimize the welfare
loss function.
The Taylor principle
In most constellations we have h > 0 as a “good” choice of weight on the
inflation term in the Taylor-rule.
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.
Therefore the real interest rate increases in the period of the “inflation shock”.
29
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 and “fiscal policy rules” (Norway for example).
There are good reasons for this: reduce influence of pressure groups, a stable
framework for policy; transparency, tie oneself to the mast (Ulysses and the
sirens), and time consistent policy (which we will review in the next lecture).
30