Download Stabilisation policy under Romer IS-MP-IA

EC201: Worksheet 4 with Romer model
Boromeus Wanengkirtyo
Department of Economics
University of Warwick
1
Introduction to IS-MP
The Romer model consists of three equations:
MP :
IS :
rt = a1 + φπ (πt − π ? ) + φy (yt − y N )
rt = c − d · yt
I A : π t = π t −1 + f ( y t −1 − y N ) + ε t
MP is the Taylor rule, which describes the interest-rate policy rule that the
central bank (CB) follows. The parameter φπ describes the response of the CB to
deviations of inflation from the target inflation rate. Likewise, φy is the response
of the CB to output gap fluctuations. a1 contains the natural rate of interest (the
interest rate the CB sets when there is no inflation or output gap deviations), as
well as a discretionary term. The CB can change a1 if it wants to deviate away
from the Taylor rule prescribed interest rate (‘discretionary policy’). Note that as
we plot the MP curve on r-Y space, the curve is for a given level of inflation. If
inflation changes, the MP curve will shift.
The IS curve describes the demand-side (consumption, investment etc). 1/b is
the interest elasticity of output. Therefore, there is a negative relationship between
consumption and investment demand, with interest rates. We use the MP and IS
curve to derive the AD curve.
Finally, the supply side of the economy is modelled with the IA (inflation
adjustment) curve. In the Romer model, inflation is pre-determined (i.e. it is only
affected by last period’s output gap and inflation, and not current output gap).
Therefore, it is a flat curve since we are plotting the IA curve on πt − yt space. ε t
is shocks to price setting (‘inflation shocks’). I explain this in more detail later.
See Figure 1. Diagrammatically, to plot the AD curve, simply take some inflation π1 and find the equilibrium output in the IS-MP graph (since the MP curve is
for a given rate of inflation). Then take a higher inflation point π2 . This shifts up
the MP curve as the central bank raises rates endogenously (i.e. autonomously).
This gets a lower equilibrium output, so we end up with a downward sloping
AD curve. The emphasis on endogenous movements of interest rates is crucial.
1
A shift of the MP curve due to inflation changes causes a movement along the AD
curve. There could be other things that shift the MP curve (for example, a1 , π ? ,
y N etc.) These things will not only shift the MP curve, but also shift the AD curve
too. This is because for a given inflation, there is a different equilibrium output.
Therefore, a change in the CB’s discretionary policy (a1 ) will cause a shift in the
AD curve.
To get the AD curve mathematically, eliminate the variable rt from the IS and
MP equations (i.e. equate them), to get:
πt = π ? +
i
1 h
c + φy y N − a1 − (b + φy )yt
φπ
The slope of the AD curve is therefore −(b + φy )/φπ .
2
Q5: Adverse Demand Shocks
See Figure 2. An adverse demand shock could be the example of a fall in consumer
confidence, like in Romer’s paper. The effect on impact is a shift of IS to the left (a
fall in the parameter c of the IS curve). This will also shift the AD curve to the left,
since for a given inflation rate, there is lower equilibrium output as determined by
the IS-MP graph. Therefore, we jump from point (1) to point (2). What happens
afterwards depends on the behaviour of the CB.
2.1
Rule-based policy
One possibility is that the CB adheres to the Taylor rule. At point (2) it will set
a lower interest rate endogenously, due to the lower output. Note that this is a
movement along the MP curve, because we are changing output. At this point,
there is a recession (negative output gap, y < y N ), so the IA curve starts shifting
down and inflation falls. This will continue to happen until the output reaches y N .
At the same time, because inflation falls, the MP curve also shifts down. Therefore,
the long-run equilibrium under this monetary policy decision is a return to y N , but
at a lower level of inflation.
2.2
Discretionary policy
The above policy is suboptimal. The economy undergoes a recession, and in the
long-run, inflation is below target (exposing the CB to possible zero lower bound
issues). The CB can instead intervene directly if it recognises the IS shock quickly.
2
This is an exogenous change in the Taylor rule (a1 ). This means that the MP
curve shifts down straight to (r3 , y N ). The AD curve shifts back to the original
position. Therefore, the economy does not suffer from a recession while inflation
is on target.
2.3
Divine Coincidence
This is the ‘divine coincidence’ result — that is, in the face of demand shocks,
when the central bank stabilises inflation, it stabilises output too. A way to see
the divine coincidence result mathematically is to have a strict inflation targeting
central bank (i.e. φπ → ∞). In other words, the central bank responds very
aggressively to any inflation deviations, to keep to its inflation target. This implies
that the slope of the AD curve is zero, meaning that demand shocks (changes in
c) has no effect on output and inflation (a leftward shift of a horizontal AD curve
is still the same AD curve.
3
3.1
Q6: Transitory adverse supply shocks
Definition of supply shocks
Firstly, we have to clarify that there are to types of supply-side shocks that the
economy faces:
• Inflation shocks ε t — shocks to price-setting
• Supply shocks — shocks to y N
The first causes shifts in the IA curve on impact. Oil price shocks would be an
example of an inflation shock. The second causes changes in y N . In turn, this may
shift the IA curve slowly if there are output deviations from the natural rate.
3.2
Inflation shocks
See Figure 3. I take at this point the interpretation a temporary inflation shock.
Therefore, the IA curve jumps up, and we go from point (1) to point (2). There is
higher inflation, π2 , so the MP curve shifts up. Note that this is the endogenous
response of the Taylor rule to higher inflation, so this is merely a movement along
the AD curve. There is a recession, so this puts downward pressure on the IA
curve, so in the long-run, we return to point (1) at (y N , π1 ).
3
3.3
Inflation-output tradeoff
What if the CB intervenes and changes a1 ? Now there is no divine coincidence,
the CB chooses between stabilising output or stabilising inflation (or somewhere
in the middle). If it chooses to stablise output, it can do so by shifting the MP
curve at the same time as the inflation shock, boosting output. This shifts the AD
curve to the right, so we end up at point (3). There is no recession, as y = y N ,
but the cost is high inflation. Another possibility is inflation stabilisation. The CB
can raise interest rates instead, creating a large recession. But in return, the IA
curve shifts down rapidly so it returns to the low-inflation outcome more quickly
at point (1).
4
Q8: Permanent adverse supply shock (doves vs hawks)
Firstly, we need to clarify the properties of a dovish and hawkish policymaker.
A dove cares about stabilising output (large φy , small φπ ), while a hawk cares
more about stabilising inflation (small φy , large φπ ). There is no perfect mapping
between the CB’s preferences over inflation and output and the parameters as
the Taylor rule is only an ad-hoc, reduced-form, way of modelling CB behaviour.
Later, I show mathematically that the source of dovishness (φy or φπ ) matters for
the equilibrium outcome.
However, the optimal (discretionary) policy for both dovish and hawkish policymakers is the same. In addition to the endogenous rise in interest rates from the
fall in y N , the CB should contract monetary policy even more to ensure that y falls
directly to y N . Therefore, there will be no inflationary pressure on the IA curve,
and output is at its new lower natural level. I show below that the endogenous
response of interest rates is never enough to fully reduce output to y N , no matter
the parameterisation of the Taylor rule.
4.1
Dynamics
What I analyse now is the dynamics of a rules-based policy, comparing a hawkish
and dovish stance. The fall in potential output creates more complex dynamics as
now there are two effects, and requires mathematics to quantify the two effects.
We want to see what parameters determine the long-run rise in inflation from this
shock. See Figure 4.
4
4.1.1
Shift of AD curve
The first effect is a shift of the MP curve, and therefore, a leftward shift of the AD
curve. Rearrange the AD curve to make y the subject:
yt =
i
1 h
φπ
c + φy y N − a1 −
(πt − π ? )
b + φy
b + φy
Given that πt is a given (i.e. flat IA line), we can take it as a constant for the
moment. Thus, if y N falls by 1 unit, yt falls by φy /(b + φy ). This is the horizontal
movement along the IA curve. But given that φy ≥ 0 and b > 0, this number is
always less than 1. Therefore, it must be the case that after the shock, y > y N .
Obviously, the number is equal to zero if φy = 0 (i.e. there is no shift in the AD
curve).
4.1.2
Slope of AD curve
After the shock, there are no more shifts of the AD curve as all the other parameters remain constant. The second effect is the adjustment occurs upwards along
the AD curve, due to inflationary pressure on the IA curve from y > y N . This adjustment stops when output equals its natural rate. What determines the inflation
rise from this adjustment is the slope of the AD curve: −(b + φy )/φπ .
4.2
Overall effects
Quantifying the inflation increase is easy, because it is just the height of the triangle. The distance (y − y N ) that the economy has to adjust until it reaches the
higher inflation equilibrium is the difference between the fall in y N (1 unit) and
the jump:
1−
φy
b
=
b + φy
b + φy
The height is just the distance covered times the slope of the AD curve:
∆π =
b + φy
b
b
·
=
b + φy
φπ
φπ
Thus, we arrive at a result: the long-run inflation increase does not depend on the
amount of output targeting — rather, only on the response to inflation deviations.
5
r
r2
MP(π 2 )
r1
MP(π 1 )
IS
Y
π
π2
π1
IA
AD
Y
Figure 1: Derivation of AD Curve
6
r
r1
2
1
MP(π 1 )
3
r2
IS 2
MP(π 2 )
IS
1
Y
π
π1
2
1
IA 1
3
π2
IA LR
AD2
Y2
YN
Figure 2: Adverse Demand Shock
7
AD1
Y
r
r1
2
MP(π 2 )
1
r2
MP(π 1 )
IS
1
Y
π
π1
2
IA 2
1
π2
IA1
AD1
Y2
YN
Figure 3: Transitory Inflation Shock
8
Y
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Slope of AD curve
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eters remain constant.
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eters remain constant.
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3
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Quantifying
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2
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4.2 Overall
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IA
1
1
at a result: the long-run inflation increase does not depend on the
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the jump:
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5
higher
inflation equilibrium
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the jump:
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y fp
y
p
The height
is just theinflation
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curve:
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result:
thearrive
long-run
not
depend
onthe
theAD
we
at
a result:
the long-run
inflation
increase
does
not depend on the
Figure
4: Permanent
Supply
Shock
amount of output targeting
on the
to inflation
deviations.
bon
+ the
fy response
amount —
of rather,
output only
targeting
—response
rather,b only
to inflation deviations.
b
Dp =
·
=
b + fy
fp
fp
5
5
Thus, we arrive at a result: the long-run inflation increase does not depend on the
amount of output targeting — rather, only on the response to inflation deviations.
5
9