Macroeconomics Theory 1
ECO – 2201
Instructor – Piyali Banerjee
Monsoon Semester
Chapter 3: The 3-Equation Model and Macroeconomic
Policy
1
•
In this chapter we will discuss about 3-equation model, which comes from
the IS-curve (the demand side of the economy), Phillips curves (PC, the
supply side of the economy), and the Monetary rule (MR, adopted by the
Central Bank to stabilize the inflation of the country).
•
At the end of this chapter, you will learn:

The role of the central bank in inflation targeting

The Monetary Rule (MR) curve

The 3-equation model (IS-PC-MR)

How inflation is stabilized in the 3-equation model
1. Role of Central Bank in Stabilizing the Demand:
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2. Inflation Targeting and The Central Bank (CB):
Question: Why does Central Bank play the main role in controlling inflation?

The natural rate of unemployment is determined by the supply-side (Ch. 2);
The CB lacks supply-side policy instruments, and thus is not the right policy
maker to reduce equilibrium unemployment.
Question: Why is inflation a concern to governments?

Electoral Motivation.
Question: Why do governments hand over inflation targeting 1 to the CB?

Monetary Policy is a preferred stabilization tool: Politically-neutral; Changes
to Fiscal Policy require lengthy parliamentary processes

Monetary Policy is delegated to an independent central bank: Governments
have low inflation credibility since there is political motivation to reduce
unemployment by manipulating MP.
Inflation targeting is a monetary policy where a central bank follows an explicit
target for the inflation rate for the medium-term and announces this inflation target to
the public. The assumption is that the best that monetary policy can do to support longterm growth of the economy is to maintain price stability, and price stability is achieved
by controlling inflation. The central bank uses interest rates, its main short-term
monetary instrument.
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An inflation-targeting central bank will raise or lower interest rates based on abovetarget or below-target inflation, respectively. The conventional wisdom is that raising
interest rates usually cools the economy to rein in inflation; lowering interest rates
usually accelerates the economy, thereby boosting inflation. The first three countries to
implement fully-fledged inflation targeting were New Zealand, Canada and the United
Kingdom in the early 1990s, although Germany had adopted many elements of inflation
targeting earlier.
In India, the Central Government has notified in the Official Gazette 4 per cent
Consumer Price Index (CPI) inflation as the target for the period from August 5, 2016
to March 31, 2021 with the upper tolerance limit of 6 per cent and the lower tolerance
limit of 2 per cent.
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
An independent CB is free from political pressure. It is likelier to deliver a
low and stable level of inflation.
Objective of Stabilization Policy:
1. To maintain a low targeted rate of inflation.
2. To have an inflation targeting monetary regime.
e.g.
i) In response to high inflation in medium-run:

CB ↑ the interest rate

Investment spending ↓

Sellers end up with unsold inventories



Production of the economy (y) ↓ ⇒ Average income of the country ↓
Price level of the country ↓
Cost of increasing interest rate: ↓ in output growth and ↑ in unemployment.
ii) In response to deflation in medium-run:

CB ↓ the interest rate

Investment spending ↑

More consumption spending on goods and services



Production of the economy (y) ↑ ⇒ Average income of the country ↑
Price level of the country ↑
Cost of decreasing interest rate: 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖.
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Adoption of Inflation Targeting: Lower and more stable inflation.
Figure 1. Inflation rates: before and after adopting the inflation targeting: 1980 to 2010
Source: IMF world economic outlook, September 2011
Question: When CB can’t stabilize the inflation with monetary policy?
i)
When there is high negative demand shock that even CB cuts the
nominal interest rate all the way to zero, which is not enough to
stimulate AD ⇒ stagflation. 2
Stagflation is persistent high inflation combined with high unemployment and stagnant
demand in a country’s economy.
2
Causes of stagflation includes cost push inflation, which occurs when some force or
condition increases the cost of production, e.g. government policies such as taxes,
shortage of natural resources etc.
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ii)
When the economy has fixed exchange rate. E.g. common currency
Euro – countries using euro can’t have their own monetary policy to
stabilize the shocks in AD. They have to use their own country-specific
fiscal policies.
Consequences of volatile inflation:
Question: Why rising inflation is not good for any economy?
•
Rising inflation reflects a distributional conflict between firms and wagesetters: Low U (unemployment)  ↑ wage-setting power  ↑ wages  ↑
inflation.
•
Rising inflation is usually followed by painful periods of disinflation.
E.g. UK in the 1980s: Disinflationary process involved a period of high
unemployment.
•
Rising inflation reduces the standard of living.
•
Increasing inflation leads to persistent frustration of workers’ real wage
expectation.
Figure 2. U.K. inflation and unemployment: 1971 – 2010
Source: UK office of National Statistics
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Low and stable inflation is beneficial:
•
Volatile inflation distorts resource allocation as it blurs any changes in
relative prices, thus leading to inefficient allocations.
•
Shoe-leather costs and menu cost inefficiencies, etc.
Question: But why target a positive inflation rate (i.e. not 0%)?
The dangers of deflation:
i)
The deflationary trap: The nominal interest rate hits the zero-bound
Weak demand ↓ inflation  ↓ real interest rate  ↓ demand  …
ii)
Increased real value of debt burdens:
Deflation increases debt as a proportion of income.
E.g. – You borrow Rs. 50,000 from your friend, which can buy 10 gm of 24 carat today. After a
month, there is a deflation, and the 24-carat gold price becomes Rs. 46,000 per 10 gm. Now,
when you repay the money (R. 50,000) to your friend after a month, you are losing Rs. 4000;
which is the worth the value of the gold.
iii)
Resistance to nominal wage cuts:
Inflation makes it easier to reduce real wages when necessary.
3. Central Bank’s Objective:
Question 1: What is the CB trying to achieve?
•
Its aim is to use monetary policy to stabilize economy and to keep the
economy close to the equilibrium output and to keep inflation at its
targeted rate.
•
The CB may be penalized if it fails to meet the inflation target.
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Question 2: What forces prevent CB from achieving its target?
•
The various shocks in aggregate demand and supply.
e.g. Boom ⇒ ↑ in AD ⇒ the positive gap between WS-PS curves and inflation
continues.
•
CB must take care of such shock and to keep the economy at a persistent
level of inflation.
Question 3: How does CB translate its objectives into monetary policy?
•
Through monetary rule (MR).
•
Taking output in the horizontal axis and inflation on vertical axis, CB’s
monetary rule chooses the preferred policy response given the constraints
it faces from the behavior of wage-setters and price-setters, which is
captured by Phillips curve.
The MR curve illustrates CB’s best response to any shock.
It then uses the relationship between interest rate and output in IS-curve to
implement its choice.
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4. The 3-equation model: IS, PC and MR Curves:
Figure 3. The 3-equation model: the adjustment of an economy to a permanent demand shock
Consumption boom:
•
IS-curve: ↑ in AD (Permanent shock) ⇒ IS-curve shifts right ⇒ No
stabilization policy through monetary rule (MR), so interest rate stays
the same, but output increases from equilibrium to high output.
•
WS-PS curves: Positive gap (1%) between WS and PS curves ⇒ wage-
setters would expect to increase their wage by 1% in the first round ⇒
price setters would immediately increase the price by 1% to keep the
profit margin same as before.
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•
MR-curve:

At the first round, CB would try to cut the inflation.

CB’s optimal response is to keep interest rate at C. At point-C,
output falls below the equilibrium output.

A negative gap (0.5%) between WS and PS curves and wage
setters would expect their wage to fall.

Price will immediately drop by 0.5%.

The central bank will choose point D as its best response to the
new inflation environment it faces and will adjust the interest
rate down from its peak.

In the subsequent periods, the central bank will continue to
guide the economy along the MR curve by reducing the interest
rate as inflation returns to the target rate.

The adjustment process finishes when the economy is back at
equilibrium output and target inflation (point Z).
Note: There will, however, be a higher interest rate at the new equilibrium, as
shown in the is diagram (the top panel of Fig. 3). A reduced level of investment
is needed to offset the persistent consumption boom.
5. Deriving the Monetary Rule (MR):
5.1.
Define the central bank’s preferences in terms of deviations from
inflation target and equilibrium output.

Define the central bank’s preferences in terms of a utility (or loss) function
to capture the costs it incurs of being away from the inflation target and
from equilibrium output. This produces the policy maker's indifference
curves in output-inflation space and shows what the policy maker would
like to do—i.e. to be close to the inflation target at equilibrium output.
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Step 1.
a) CB’s objective to bring the inflation in the targeted level: 𝜋𝜋 𝑇𝑇
b) At period-t, if the inflation is 𝜋𝜋𝑡𝑡 , CB tries to minimize the inflation gap
which is: (𝜋𝜋𝑡𝑡 − 𝜋𝜋 𝑇𝑇 )2
← 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣.
c) CB’s second concern is to reduce the output gap from its equilibrium at
period t, which is denoted by:
Step 2.
(𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑒𝑒 )2
← 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣.
The sum of inflation and output gap is CB’s total utility loss at the given period
t, denoted by:
𝐿𝐿 = (𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑒𝑒 )2 + 𝛽𝛽(𝜋𝜋𝑡𝑡 − 𝜋𝜋 𝑇𝑇 )2
(1)
where, 𝛽𝛽 is the relative weight attached to the loss from inflation. This is a critical
parameter:
a) 𝛽𝛽 > 1 ⇒ central bank that places less weight on deviations in employment
from its target than on deviations in inflation, and vice versa. A more
inflation—averse central bank is characterized by a higher 𝛽𝛽. This makes
the indifference curves ellipsoid as in Fig. 4b. They are flat because the
central bank is willing to trade off a small fall in inflation for a large rise in
unemployment above equilibrium.
b) 𝛽𝛽 = 1 ⇒ the central bank is equally concerned about inflation and output
deviations from its targets. The loss function is simple to draw: each
indifference curve is a circle with (𝑦𝑦𝑒𝑒 , 𝜋𝜋 𝑇𝑇 ) at its centre (see Fig. 4a). The loss
declines as the circle gets smaller. When 𝜋𝜋𝑡𝑡 = 𝜋𝜋 𝑇𝑇 and 𝑦𝑦𝑡𝑡 = 𝑦𝑦𝑒𝑒 , the circle
shrinks to a single point (called the 'bliss point') and the loss is at a
minimum, which is zero. With 𝛽𝛽 = 1, the central bank is indifferent
between inflation 1% above (or below) 𝜋𝜋 𝑇𝑇 and output 1% below (or above)
ye. They are on the same loss circle.
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c) 𝛽𝛽 < 1 ⇒ central bank with less aversion to inflation and will have ellipsoid
indifference curves with a vertical rather than a horizontal orientation (Fig.
4c). In that case, the indifference curves are steep, reflecting that the
central bank is only willing to trade off a given all in inflation for a smaller
fall in output than in the other two cases.
Figure 4. Central Bank’s loss functions: utility declines with distance from the “bliss-point” when 𝜋𝜋𝑡𝑡 = 𝜋𝜋 𝑇𝑇 and 𝑦𝑦𝑡𝑡 = 𝑦𝑦𝑒𝑒 , the circle shrinks to a single point.
5.2.
Define the central bank’s constraints from the supply side, i.e., the
Phillips Curve (PC)


The Phillips curve constraint faced by the central bank:
𝜋𝜋𝑡𝑡 = 𝜋𝜋𝑡𝑡−1 + 𝛼𝛼(𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑒𝑒 )
(Adaptive expectations PC, 𝜋𝜋𝑡𝑡𝐸𝐸 = 𝜋𝜋𝑡𝑡−1 )
(2)
The PC shows combinations of output and inflation which are
attainable by the central bank, for a given inflation expectation.
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
This is shown in Fig. 5, where the upward sloping lines are the
Phillips curves, we worked with in Chapter 2.


In the diagram, it is assumed that 𝛼𝛼 = 1, so that each Phillips curve
has a slope of 450 .
Each Phillips curve is labelled by a given level of expected inflation,
which in the adaptive expectations Phillips curve is equal to lagged
inflation. Assume that 𝜋𝜋𝑡𝑡𝐸𝐸 = 𝜋𝜋𝑡𝑡−1 = 𝜋𝜋 𝑇𝑇 = 2 (remember that this PC

must go through point A at which 𝑦𝑦𝑡𝑡 = 𝑦𝑦𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎 𝜋𝜋 = 2).
The central bank is in the happy position of being able to choose
bliss point A or (𝜋𝜋 𝑇𝑇 , 𝑦𝑦𝑒𝑒 ) at which its loss is zero.
Question: What happens if there has been a shock to inflation and it is not equal
to the inflation target?
•
•
If the economy is on the PC of 𝜋𝜋𝑡𝑡𝐸𝐸 =4, the ‘bliss point’ A (where 𝜋𝜋𝑡𝑡 = 𝜋𝜋 𝑇𝑇 and
𝑦𝑦𝑒𝑒 = 𝑦𝑦𝑡𝑡 , and thus 𝐿𝐿 = 0 ) cannot be attained.
Trade off between two options by CB:
i)
ii)
If it wants a level of output 𝑦𝑦𝑒𝑒 next period, then it has to accept an inflation
rate above its target, i.e. 𝜋𝜋𝑡𝑡 = 4 ≠ 𝜋𝜋 𝑇𝑇 (point B in Fig. 5).
If it wishes to hit the inflation target next period, it must accept a much
lower level of output next period (point C).
Note: Point B corresponds to a fully accommodating monetary policy in which
the objective is purely to hit the output target (𝛽𝛽 = 0), and point C corresponds
to a completely non-accommodating policy, in which the objective is purely to hit
the inflation target.
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Figure 5. Loss circles and Phillips curve
5.3.
Derive the best response monetary rule in the output-inflation space,
which gives the MR curve.
•
The MR curve shows the central bank's preferred output-inflation
combination for any Phillips curve it faces.
•
It can be derived graphically by finding the points of tangency between the
Phillips curves and the loss circles.
•
As shown in Fig. 6a, points A, B and C all minimize the loss function of
the central bank for their given Phillips curve.
•
Any deviation from the tangency points results in a higher (than necessary)
loss for the CB and is hence not optimal.
•
Joining up the tangency points gives us the best-response MR curve,
which is a result of the CB minimizing its loss function (see Fig 6b).
•
The MR curve is given by: (𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑒𝑒 ) = −𝛼𝛼𝛼𝛼 (𝜋𝜋𝑡𝑡 − 𝜋𝜋 𝑇𝑇 )
Proof:
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•
It tells the CB the best-response output gap
inflation is away from target (𝜋𝜋𝑡𝑡 − 𝜋𝜋 𝑇𝑇 ≠ 0).
(𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑒𝑒 )
to choose when
Figure 6. Deriving the MR curve
5.4.
Now that the best response output gap (𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑒𝑒 ) is chosen, y is set using
the real interest rate (r) on the IS curve:
(IS curve, see Ch. 1)
•
The CB sets the nominal interest rate to achieve the desired real
interest rate (the Taylor Principle)
•
To take into account the delayed impact of interest rates on output,
the 3-equation model uses the ‘dynamic’ IS curve, given by:
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Quick Summary: How does a CB respond to an inflation shock in our 3-equation
model?
Figure 7. Dynamic adjustment of an inflation shock
Step-by-step analysis: How does a CB respond to an inflation shock?
•
Positive demand shock occurs in the economy.
Note: CB anticipates it as a temporary AD shock.
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Period 0:
•
The economy is initially at point ‘A’ (the Bliss point)
•
Inflation shock: IS shifts upwards (Economy moves to point ‘B’)
•
•
The CB forecasts PC for the next period, which is PC (𝜋𝜋1𝐸𝐸 = 𝜋𝜋0 )
•
where the MR = PC); To achieve this, 𝑟𝑟0 is set (see IS curve)
The CB chooses position on this PC which minimizes its loss (point ‘C’
𝑟𝑟 affects 𝑦𝑦 with a 1-period lag, so the economy ends with (‘𝜋𝜋0 , 𝑦𝑦𝑒𝑒 and 𝑟𝑟0 ’).
Period 1:
•
The higher 𝑟𝑟0 reduces output below equilibrium to 𝑦𝑦1 by ↓ Investment
(Economy moves to point ‘C’)
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•
•
Again, the CB forecasts PC for period 2, i.e. PC (𝜋𝜋2𝐸𝐸 = 𝜋𝜋1 ); Note that the PC
shifts as 𝜋𝜋 𝐸𝐸 is updated.
The optimal point as given by the MR is now ‘D’; To achieve this, 𝑟𝑟0 is
reduced to 𝑟𝑟1. Similarly, the economy ends with (‘𝜋𝜋1 , 𝑦𝑦1 and 𝑟𝑟1’).
Period 2 onwards:
•
The lower 𝑟𝑟1 increases output to 𝑦𝑦2 , while inflation falls to 𝜋𝜋2 (Economy
moves to point ‘D’)
•
The same process repeats until the economy is back at its equilibrium ‘Z’
•
The economy moves down along MR curve: The CB gradually adjusts 𝑟𝑟1
down to 𝑟𝑟𝑆𝑆 , output rises slowly from 𝑦𝑦2 to 𝑦𝑦𝑒𝑒 and inflation eventually falls
from 𝜋𝜋2 to 𝜋𝜋 𝑇𝑇 .
Note: Since the CB anticipates the AD shock as temporary, there will be no change
in the interest rate, even after the shock (CB operates at old IS curve only).
 Definition: The Impulse Response Function
•
A function which shows the time path of macroeconomic variables
following a shock and the resulting policy responses.
•
(see Fig 8b.) Note that the impulse response of output only changes one
period after the inflation shock. This is due to the 1-period lagged effect of
𝑟𝑟 on 𝑦𝑦.
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Figure 8. Adjustment to an Inflation Shock and The Impulse Reponses
Exercise 1. Show the adjustment process through graphs in 3-equation model
when there is a positive and permanent aggregate supply shock.
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Summary:

The 3-equation model consists of the following:

The IS curve models the demand side, the PC curve models the supply side
and the MR curve models the policy maker.

Stabilization policy: Low & stable inflation minimizes the negative effects
of inflation on the economy.

Dangers of the deflation trap discourage policy makers from targeting a
0% inflation rate.

Achieving the CB’s objectives:
o The Monetary Rule (MR) shows the output and inflation combinations
which minimize the CB’s loss function.
o The PC forms a constraint to the CB’s choice of output gap.
o The desired output gap is implemented by the CB using the int. rate (𝑟𝑟).
This is done following the IS relation.
o Demand shocks cause a deviation of output from equilibrium while supply
shocks move the economy to a new equilibrium.

Role of lags & forecasting
o The real int. rate affects AD with a time lag: reflected in the model by the
‘dynamic’ IS curve.
o The time lag means that forecasting (e.g. the PC curve and the IS curve) is
vital for the process of setting the int. rate.
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