06/06/2026, 16:16 ECON2291 — Three Equation Model & Inflation Bias
ECON2291 · Economic Theory · Revision Guide
The Three Equation Model, Monetary
Policy & Inflation Bias
COMPREHENSIVE REVISION GUIDE — ECON2291-WE01
CONTENTS
Part I — The Three Equation Model
1. Overview and motivation
2. The IS curve
3. The Phillips curve (IAPC)
4. The monetary rule — derivation
5. Equilibrium and the two-quadrant diagram
6. Shock analysis
7. The double lag and the Taylor rule
Part II — Inflation Bias & Time Inconsistency
8. Why governments inflate
9. Inflation bias under adaptive expectations
10. Inflation bias under rational expectations
11. Time inconsistency (Kydland & Prescott 1977)
12. Solutions — rules and central bank independence
13. Alesina & Summers (1993) — empirical evidence
Part III — Past Paper Questions
Part IV — Practice Questions
PA R T O N E
The Three Equation Model
1. Overview and Motivation
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,06/06/2026, 16:16 ECON2291 — Three Equation Model & Inflation Bias
The three equation model replaces the IS-LM framework by recognising that modern central banks
set the interest rate rather than targeting the money supply. Instead of an LM curve derived from
money market equilibrium, the model contains a monetary rule (MR) derived from an explicit
optimisation of the central bank's loss function. The result is a framework considerably better
suited to understanding inflation targeting regimes.
THE THREE EQUATIONS
The model consists of three schedules operating jointly:
IS curve — in (r, y) space: output demanded as a function of the real interest rate
Phillips curve (IAPC) — in (π, y) space: current inflation as a function of past inflation
and the output gap
Monetary rule (MR) — in (π, y) space: the central bank's optimal choice of output
given every possible Phillips curve
The model is displayed graphically across two quadrants. The upper quadrant plots the IS curve in
(y, r) space. The lower quadrant plots both the IAPC and the MR in (y, π) space. A vertical dashed
line at y = ye connects the two, so the interest rate the central bank sets can be read off the IS curve
once the target output is determined below.
2. The IS Curve
IS CURVE
yt = A − art−1
yt A
Output (log) in period t Autonomous component: consumer and
investor confidence, government expenditure
a rt-1
Interest sensitivity of aggregate demand Real interest rate set in the previous period
(positive)
Three features demand attention. First, there is a one-period lag: the interest rate set today
affects output only next period. This is a stylised reflection of the time required for monetary policy
to feed through credit markets into spending decisions. Second, the relationship is negative: higher
real interest rates reduce borrowing and investment, contracting output. Third, the curve sits in (r,
file:///C:/Users/tyala/Downloads/econ2291_three_eq_inflation_guide (1).html 2/31
,06/06/2026, 16:16 ECON2291 — Three Equation Model & Inflation Bias
y) space and is downward sloping — the counterpart to the familiar IS curve, but with r rather than
the nominal rate on the axis.
The stabilising rate rS is defined as the level of r at which output equals its equilibrium level ye.
This is the real interest rate consistent with stable inflation at the target.
WHY NOT USE THE LM?
The IS-LM model's LM curve requires stable money demand, which became empirically
questionable during the financial liberalisation of the 1980s. Central banks found it more
reliable to target short-term interest rates directly. The three equation model makes this the
behavioural assumption rather than a special case.
3. The Phillips Curve — IAPC
INERTIA-AUGMENTED PHILLIPS CURVE ( I APC)
πt = πt−1 + α(yt − ye)
This is a version of Friedman's (1968) expectations-augmented Phillips curve. Inflation is
determined by last period's inflation (which proxies expected inflation) plus a term capturing
demand pressure: how far output departs from its equilibrium level. When output exceeds ye,
falling unemployment puts upward pressure on wages and prices, raising inflation above its
inherited level.
Why use lagged inflation as the expectation?
Expectations are stated as πe = πt-1. This does not imply agents are naive or that they have adaptive
expectations in the strict sense. Rather, it captures the institutional reality that firms and
households try to compensate for actual past inflation in wage bargaining and price-setting.
Contracts written today index future nominal values to past price levels. The justification is partly
institutional and partly empirical — the model fits the data well and is pedagogically tractable.
πt πt-1
Current inflation rate Last period's inflation (= expected inflation)
α ye
Responsiveness of inflation to output gap Equilibrium (full-employment) output
(positive)
file:///C:/Users/tyala/Downloads/econ2291_three_eq_inflation_guide (1).html 3/31
, 06/06/2026, 16:16 ECON2291 — Three Equation Model & Inflation Bias
In (y, π) space, the IAPC is upward sloping with slope α and intercept πt-1. A shift in last period's
inflation shifts the entire curve vertically. A positive output gap moves the economy along a given
IAPC to a higher inflation rate; the next period's IAPC then shifts up by the same amount,
embedding the higher inflation into expectations.
⚠ Exam note: The IAPC is not the same as adaptive expectations. The model does not claim agents
form πe = πt-1 because they extrapolate mechanically from the past. It claims they bargain to
recover past inflation losses. This distinction matters if an exam question asks you to distinguish
the IAPC from a purely adaptive expectations model.
4. The Monetary Rule — Derivation
The monetary rule is not assumed — it is derived from an explicit optimisation. The central bank
minimises a loss function that penalises deviations of output from ye and deviations of inflation
from the target πT.
CENTRAL BANK LOSS FUNCTION
2 T 2
L = (yt − ye) + β(πt − π )
β ≥ 0 captures the central bank's relative aversion to inflation deviations. Higher β → more
inflation-averse bank.
DERIVATION OF THE MONETARY RULE
STEP-BY-STEP PROOF
1 Substitute IAPC into loss function. Replace πt in L using πt = πt-1 + α(yt −
ye):
2 T 2
L = (yt − ye) + β(πt−1 + α(yt − ye) − π )
2 Differentiate with respect to yt and set equal to zero (first-order
condition):
dL T
= 2(yt − ye) + 2αβ(πt−1 + α(yt − ye) − π ) = 0
dyt
Dividing through by 2:
file:///C:/Users/tyala/Downloads/econ2291_three_eq_inflation_guide (1).html 4/31
ECON2291 · Economic Theory · Revision Guide
The Three Equation Model, Monetary
Policy & Inflation Bias
COMPREHENSIVE REVISION GUIDE — ECON2291-WE01
CONTENTS
Part I — The Three Equation Model
1. Overview and motivation
2. The IS curve
3. The Phillips curve (IAPC)
4. The monetary rule — derivation
5. Equilibrium and the two-quadrant diagram
6. Shock analysis
7. The double lag and the Taylor rule
Part II — Inflation Bias & Time Inconsistency
8. Why governments inflate
9. Inflation bias under adaptive expectations
10. Inflation bias under rational expectations
11. Time inconsistency (Kydland & Prescott 1977)
12. Solutions — rules and central bank independence
13. Alesina & Summers (1993) — empirical evidence
Part III — Past Paper Questions
Part IV — Practice Questions
PA R T O N E
The Three Equation Model
1. Overview and Motivation
file:///C:/Users/tyala/Downloads/econ2291_three_eq_inflation_guide (1).html 1/31
,06/06/2026, 16:16 ECON2291 — Three Equation Model & Inflation Bias
The three equation model replaces the IS-LM framework by recognising that modern central banks
set the interest rate rather than targeting the money supply. Instead of an LM curve derived from
money market equilibrium, the model contains a monetary rule (MR) derived from an explicit
optimisation of the central bank's loss function. The result is a framework considerably better
suited to understanding inflation targeting regimes.
THE THREE EQUATIONS
The model consists of three schedules operating jointly:
IS curve — in (r, y) space: output demanded as a function of the real interest rate
Phillips curve (IAPC) — in (π, y) space: current inflation as a function of past inflation
and the output gap
Monetary rule (MR) — in (π, y) space: the central bank's optimal choice of output
given every possible Phillips curve
The model is displayed graphically across two quadrants. The upper quadrant plots the IS curve in
(y, r) space. The lower quadrant plots both the IAPC and the MR in (y, π) space. A vertical dashed
line at y = ye connects the two, so the interest rate the central bank sets can be read off the IS curve
once the target output is determined below.
2. The IS Curve
IS CURVE
yt = A − art−1
yt A
Output (log) in period t Autonomous component: consumer and
investor confidence, government expenditure
a rt-1
Interest sensitivity of aggregate demand Real interest rate set in the previous period
(positive)
Three features demand attention. First, there is a one-period lag: the interest rate set today
affects output only next period. This is a stylised reflection of the time required for monetary policy
to feed through credit markets into spending decisions. Second, the relationship is negative: higher
real interest rates reduce borrowing and investment, contracting output. Third, the curve sits in (r,
file:///C:/Users/tyala/Downloads/econ2291_three_eq_inflation_guide (1).html 2/31
,06/06/2026, 16:16 ECON2291 — Three Equation Model & Inflation Bias
y) space and is downward sloping — the counterpart to the familiar IS curve, but with r rather than
the nominal rate on the axis.
The stabilising rate rS is defined as the level of r at which output equals its equilibrium level ye.
This is the real interest rate consistent with stable inflation at the target.
WHY NOT USE THE LM?
The IS-LM model's LM curve requires stable money demand, which became empirically
questionable during the financial liberalisation of the 1980s. Central banks found it more
reliable to target short-term interest rates directly. The three equation model makes this the
behavioural assumption rather than a special case.
3. The Phillips Curve — IAPC
INERTIA-AUGMENTED PHILLIPS CURVE ( I APC)
πt = πt−1 + α(yt − ye)
This is a version of Friedman's (1968) expectations-augmented Phillips curve. Inflation is
determined by last period's inflation (which proxies expected inflation) plus a term capturing
demand pressure: how far output departs from its equilibrium level. When output exceeds ye,
falling unemployment puts upward pressure on wages and prices, raising inflation above its
inherited level.
Why use lagged inflation as the expectation?
Expectations are stated as πe = πt-1. This does not imply agents are naive or that they have adaptive
expectations in the strict sense. Rather, it captures the institutional reality that firms and
households try to compensate for actual past inflation in wage bargaining and price-setting.
Contracts written today index future nominal values to past price levels. The justification is partly
institutional and partly empirical — the model fits the data well and is pedagogically tractable.
πt πt-1
Current inflation rate Last period's inflation (= expected inflation)
α ye
Responsiveness of inflation to output gap Equilibrium (full-employment) output
(positive)
file:///C:/Users/tyala/Downloads/econ2291_three_eq_inflation_guide (1).html 3/31
, 06/06/2026, 16:16 ECON2291 — Three Equation Model & Inflation Bias
In (y, π) space, the IAPC is upward sloping with slope α and intercept πt-1. A shift in last period's
inflation shifts the entire curve vertically. A positive output gap moves the economy along a given
IAPC to a higher inflation rate; the next period's IAPC then shifts up by the same amount,
embedding the higher inflation into expectations.
⚠ Exam note: The IAPC is not the same as adaptive expectations. The model does not claim agents
form πe = πt-1 because they extrapolate mechanically from the past. It claims they bargain to
recover past inflation losses. This distinction matters if an exam question asks you to distinguish
the IAPC from a purely adaptive expectations model.
4. The Monetary Rule — Derivation
The monetary rule is not assumed — it is derived from an explicit optimisation. The central bank
minimises a loss function that penalises deviations of output from ye and deviations of inflation
from the target πT.
CENTRAL BANK LOSS FUNCTION
2 T 2
L = (yt − ye) + β(πt − π )
β ≥ 0 captures the central bank's relative aversion to inflation deviations. Higher β → more
inflation-averse bank.
DERIVATION OF THE MONETARY RULE
STEP-BY-STEP PROOF
1 Substitute IAPC into loss function. Replace πt in L using πt = πt-1 + α(yt −
ye):
2 T 2
L = (yt − ye) + β(πt−1 + α(yt − ye) − π )
2 Differentiate with respect to yt and set equal to zero (first-order
condition):
dL T
= 2(yt − ye) + 2αβ(πt−1 + α(yt − ye) − π ) = 0
dyt
Dividing through by 2:
file:///C:/Users/tyala/Downloads/econ2291_three_eq_inflation_guide (1).html 4/31
