Regime dependent macro models (cont)
From Lecture 10 we have the following system of equations:
f
yt = β0 + β1ert − β2rt + β3gt + β4yt
e
rt = it − πt+1
Lecture 11: Open economy AD-AS; fiscal
policy in different regimes.
Ragnar Nymoen
Department of Economics, University of Oslo
f
ert = ∆et + πt − πt + ert−1
πt = πte + γ(yt − ȳ) + st
(1)
(2)
(3)
(4)
f
it = it + ee + αe(∆et + et−1)
mt − pt = m0 − m1it + m2yt, mi > 0, i = 1, 2
(5)
(6)
(1) is the product marked equilibrium condition. (2) is the definition of the
e is the expected rate of inflation, one period ahead.
real interest rate. πt+1
(3) is a definition equation for ert , see IAM, p 704 and 711.
(4) is the PCM.
November 6, 2007
1
e /E ) ≡ ee
e
e
(5) is UIP with ln(Et+1
t
t+1 − et = e + α (∆et + et−1) inserted.
(6) is an equilibrium condition for the money market. Right hand side is a
linearization of the demand for money function.
2
interest rate
Short-run models
f
f f
In the short-run, the following variables are exogenous: gt, yt , st, πt , it , ert−1,
et−1 and pt. Note the discussion at the end of Lecture 10 which motivated the
Ei-curve
i1
i0
classification of pt as pre-determined (as a simplification).
The two main regimes to consider (because the are robust to perfect capital
mobility) are Regime I and Regime VI. They are different in terms of how the
interest rate is determined.
Money
M0 M1
E0 E1
exchange rate
Regime I: The shift in the Ei-curve does not affect i. E depreciates
Regme VI: To avoid depreciation, i is inceased. Accommodated in
the money market by reduction of the money supply (through market
operations)
Figure 1: Regime I and VI: FEX market and money market equilibrium
3
4
Regime I (floating ex rate)
Regime VI (fixed ex rate, it as instrument)
Regime dependent exogenous variable: mt
Regime dependent exogenous variable: ∆et.
n
f
yt = β0 + β1 ∆et + πt − πt + ert−1
n
f
o
o
(7)
f
e
+ β3gt + β4yt
− β2 it + ee + αe(∆et + et−1) − πt+1
e
πt = πt + γ(yt − ȳ) + st
f
it = it + ee + αe(∆et + et−1)
mt − pt = m0 − m1it + m2yt
Money market and FEX market is now interlocked. Solve (5) and (6) for it
and the nominal exchange rate:
1
f
(it − it − ee) − et−1
αe
−1
m
m
it =
(mt − pt) + 0 + 2 yt
m1
m1 m1
∆et =
(8)
(9)
(10)
(7) is the AD curve. Note that the equilibrium condition on the market for
e ,
foreign exchange, (9), is included in this equation. (8) is the AS curve. If πt+1
e
and πt are exogenous, then (7) and (8) determine yt and πt. it is determined
in (9) and mt in (10).
(
(
)
m
m
1 f
−1
f
(mt − pt) + 0 + 2 yt − e (it + ee) + πt − πt
m1
m1 m1
α
(
)
m0 m2
−1
e
− β2
(mt − pt) +
+
yt − πt+1
m1
m1 m1
yt = β0 + β1
1
αe
f
+ β3gt + β4yt
)
(11)
πt = πte + γ(yt − ȳ) + st
(12)
6
5
The difference between Regime I and VI is the slope of the AD curves (7) and
(11):
¯
∂πt ¯¯
1
=
<0
¯
∂yt ¯AD,rV I
−β1
(13)
¯
m1
2
1 − β1 αm
e m + β2 m
∂πt ¯¯
1
2
=
(14)
¯
∂yt ¯AD,rI
−β1
We noted that (14) hinges on αe 6= 0. The interpretation is that with constant
depreciation expectations and perfect capital mobility, it is determined by the
UIP condition alone. Hence αe = 0 would introduce an internal inconsistency
with the assumption that in this regime, mt is exogenous.
We ended Lecture 10 by the following important result about the slopes of the
short-run AS curves of the two regimes.
¯
¯
∂πt ¯¯
∂πt ¯¯
>−
, when αe < 0
−
¯
¯
¯
∂yt AD,rI
∂yt ¯AD,rV I
(15)
Interpretation of slope-difference
The interpretation of the difference has to do with how the interest rate is
determined in the two regimes: When πt increases, y-demand is reduced in
both regimes through the real exchange rate, er . But there are additional
effects in RI: Lower demand for money reduces the interest rate in the domestic
money market. Hence the eventual reduction in y-demand in RI is less than in
RVI. This is the same as saying that the slope of the AD curve is steeper in RI
than in RVI.
¯
¯
∂πt ¯¯
∂πt ¯¯
−
>−
¯
¯
∂yt ¯AD,rI
∂yt ¯AD,rV I
meaning that the slope of the short-run AD curve is steeper in Regime I than
in Regime VI, at least when αe < 0.
7
8
p
The short-run solutions for πt and yt is obtained by solving (11) and (12), for
Regime I, and (7) and (8) for Regime VI.
To learn about the properties of the two regime versions of the model we will
consider the response of the endogenous variables yt and πt.
Regime VI
For simplicity, and according to custom, we assume that the initial situation is
characterized by
πt = π e
Regime I
and
yt = ȳ,
y
e
Figure 2: Short-run AD curves, fixed πt+1
in regime I and VI.
as depicted in figure 3.
10
9
Fiscal policy in RI and RVI
p
Consider the immediate (short-run) effect of an increase in gt.
AS
p
From (11) and (7), note that for a given yt, the derivative of πt with respect
to gt is identical in the two regimes
e
¯
RVI
RI
y
y
e . Regime I and VI.
Figure 3: Initial situation with πt = πt+1
11
¯
dπt ¯¯
β
dπt ¯¯
=
= 3>0
¯
¯
¯
dgt yt=ȳ,rI
dgt ¯yt=ȳ,rI
β1
The graphical analysis of short-run effects of fiscal policy is therefore represented by identical vertical shifts in the AD curve of the two regimes.
Hence, the impact effect of increased gt is larger in Regime VI (fixed exchange
rate) than in Regime I (float), see figure 4.
The explanation is that higher GDP output increases the demand for money,
which in Regime I increases the interest rate, in regime VI the interest rate
stays constant.
12
p
AS
Long-run models
RVI
RI
pe
We still consider Regime I and VI, and repeat the equations of the model:
f
yt = β0 + β1ert − β2rt + β3gt + β4yt
(16)
ert =
(18)
rt =
y
y
πt =
it =
e
it − πt+1
f
∆et + πt − πt + ert−1
e
πt + γ(yt − ȳ) + st
f
it + ee + αe(∆et + et−1)
e
Figure 4: Short-run effects of fiscal policy, regime Regime I and VI, fixed πt+1
and πte .
mt − pt = m0 − m1it + m2yt, mi > 0, i = 1, 2
13
14
(17)
(19)
(20)
(21)
Hence, from the UIP condition (20):
The models’ steady-state is defined by the following, see IAM p. 717-719.
πte = π̄ f , expectation equal to the world inflation rate
yt = ȳ,
ert = ert−1 = er , stationarity of rex,
gt = ḡ, gov exp on trend,
f
yt = ȳ f , world GDP on trend
ı̄f = constant world interest rate
st = 0, no supply shocks
Since ert = ert−1 = er and πt = π̄ f it follows from (18) that ∆et = 0 in
e /E ) so we add
steady-state. It is logical that in a steady-state ∆et = ln(Et+1
t
e /E ) = ∆e = 0
ln(Et+1
t
t
to the list of steady-state conditions.
i = ı̄f
(22)
m − p = m0 − m1ı̄f + m2ȳ
(23)
ȳ = β0 + β1er − β2(ı̄f − π̄ f ) + β3ḡ + β4ȳ f , from AD
π = π f , from AS
(24)
(25)
and from (21)
e
= π̄ f in (16) gives
Using i = ı̄f and πt+1
The equations of the long-run model are thus: (24),(25), (22) and (23). The
endogenous variables of the long-run model are: er , π, i, m or e, and p.
Even though pt is predetermined in the short-run model it is endogenous in
the long-run. Hence there is one missing equation in our long-run model.
However, note that ert = ert−1 = er is actually the hypothesis of PPP, as a
long-run property. Hence, from the definition of er
p = e + pf − er
15
16
(26)
which determines p (noting that er is determined in the (24) and pf is exogenous.
The PPP condition (26) determines p in the Regime VI since e is exogenous
and fixed.
e /E ) = 0 which
In regime I, e is determined from the long run condition ln(Et+1
t
gives
Hence, if the increase in g is permanent, the model predicts a long-run there
real appreciation. In both regimes, we can think as if this takes place trough
increased P , i.e. using (26).
0 = ee + αee
and e in Regime 1. Hence, in the long-run model, p is determined by PPP also
in regime I. Hence, m is assumed to be endogenous in the long-run version of
regime I.
Regime independency of the long-run solution (IAM figure 23.7)
However, as advanced courses on flexible exchange rates will show, the stability
of the nominal dynamic adjustment process in regime I is not straight-forward.
The dynamics for regime VI is easier to analyze explicitly, and we do that in
the next lecture.
(24) is the same in both regimes (I and VI). The long-run AS schedule is also
identical across the two regimes. Hence the steady-state solutions for er and
π are independent of the two regimes.
¯
der ¯¯
−β3
=
<0
¯
dḡ ¯rI,rV I
β1
18
17
e
r
e
r
LR AS
LRAD
Short-summary of our analysis of the AD-AS model so far.
r
e0
y
y
Float/fix
Target (exogenous)
Instrument
short-run effects of g
long-run effects of g
RI
R VI
float
M
i
yt and πt ↑, er ↓
er ↓,
fix
E
i
larger on yt and πt
same as RI
Figure 5: Long-run effect of fiscal policy, regime Regime I and VI .
19
20
Summary of regime dependent open economy macro model so
More on fiscal policy in the fixed exchange rate regime
far
We have analyzed the short-run and long-run effects of a permanent change in
gt in two exchange rate regimes: Float with money supply as exogenous target
variable, and fixed with Et as exogenous target variable and i as instrument.
The above is based on fixed and exogenous inflation and a permanent policy
change. We now relax these assumptions, and we also want to say something
about dynamics.
a. Change in gt is permanent and πte and πte are constant
Main conclusions:
1. The regimes can be graphically represented in one graph: the slopes of the
AD lines are different. Steeper in the floating ex-rate regime
2. Short-run effect of fiscal policy expansion largest in case of fixed ex-rate
regime
3. No long-run effects on y in either regime. The new long run equilibrium
level of ert is lower than before the shock. Nominal appreciation in the case
of float. Increased P in the fixed exchange rate regime.
Short-run: AD curve shifts. Dynamics: AD curve shift gradually back again.
e
The economy is “gliding” back to initial equilibrium point (πt+1
= πte = π f
r
and yt = ȳ). Gradually et is reduced.
e
b. As in case a) but inflation expectations are rational in the sense of πt+1
=
e
f
πt = π , see page 739 in IAM.
Same analysis!
21
Formal analysis of dynamics and stability
22
The model can be written in somewhat more compact form as
f
yt − ȳ = β̂0 − β1(πt − πt ) + β1ert−1 + β3gt + dt,
f
f
f
dt = −β2(it − πt ) + β4yt − ȳ
See 739-741 in ch 24.3 of IAM.
f
So far we have implitely assumed that the dynamic process which is triggered
by the raise in gt is stable, so that a new long-run equilibrium is reached.
Intuitively, the dynamics are in fact stable, since a process of real currency
appreciation is begins as a result of the shock. As an exercise, this can be
shown formally.
πt − πt = γ(yt − ȳ) + st
f
ert = −(πt − πt ) + ert−1
Use (27) and (28) to express for ert−1 by yt − ȳ (e.g.
f
(πt − πt )):
ert−1 =
Fixed ex-rate assumptions:
(27)
(28)
(29)
by substitution for
1
β
dt
β̂
(1 + β1γ)(yt − ȳ) + st − 3 gt −
− 0
β1
β1
β1 β1
which is equation (17) page 739. We also have:
e
πt+1
= πte = π f
eet+1 − et = 0
∆et = 0
23
ert =
1
β
d
β̂
(1 + β1γ)(yt+1 − ȳ) + st+1 − 3 gt+1 − t+1 − 0
β1
β1
β1
β1
Using these two expressions in (29), together with (28), gives the following
24
ADL model for (yt+1 − ȳ):
(yt+1 − ȳ) =
1
β3
(yt − ȳ) +
∆gt+1 + ....
1 + β1γ
(1 + β1γ)
Temporary versus permanent fiscal policy shocks (IAM p 24.3)
or, for yt :
(30)
We can also use (30) to analyze the different responses to permanent and
temporary fiscal policy shocks.
We can now apply what we have learnt about stability earlier, namely that
(yt − ȳ) is dynamically stable if
A permanent shock amounts to ∆g1 > 0, for example ∆g1 = 1, and ∆g2 =
∆g3 = ... = 0.
(yt − ȳ) =
1
β3
(yt−1 − ȳ) +
∆gt + ....
1 + β1γ
(1 + β1γ)
1
<1
1 + β1γ
A temporary shock amounts to ∆g1 = 1, ∆g2 = −1, for example, and ∆g3 =
which holds since β1γ > 0.
25
26
∆g4 = ... = 0.
perm
temp
impact
β3
(1+β1γ)
β3
(1+β1γ)
2nd
β3
(1+β1γ)2
−β1γβ3
(1+β1γ)2
Intuitively, in this model, systematic fiscal policy leads to a steeper short-run
AD schedule. A real appreciation leads to less reduction in y in the case of
a > 0, than a = 0, see fig 24.4.
3rd
β3
(1+β1γ)3
−β1γβ3
(1+β1γ)3
Effects of negative demand shocks are smothered.
...
0
...
0
The speed of adjustment is reduced.
Hence the temporary leads to a short expansion, and then a longer period where
yt − ȳ < 0.
So far we have considered unsystematic fiscal policy, An example of systematic
fiscal policy is:
gt − ḡ = a(ȳ − yt),
27
a > 0.
(31)
28