Advanced Macroeconomics
Tutorial: Optimal Monetary Policy Rule in the LQ Model
Ilya Gulenkov
Higher School of Economics
Autumn 2023
Contact: igulenkov@hse.ru
Ilya Gulenkov (Higher School of Economics)
Advanced Macroeconomics
Autumn 2023
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Description of the Economy
1
Loss function: L = (Yt − Y ∗ )2 + α(πt − π̄)2
2
e
Phillips curve: πt = πt + λ(Yt − Yn )
IS curve: Yt = multA (A − brt )
Y ∗ is the level of output targeted by the CB
π̄t is the inflation target
πte is the level of inflation expectations
Yn is the natural level of output consistent with the labour market equilibrium
α is the relative weight of inflation stabilization in the CB’s preferences
λ is the sensitivity of inflation to output gap
Ilya Gulenkov (Higher School of Economics)
Advanced Macroeconomics
Autumn 2023
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Deriving the Monetary Policy (MP) Curve
Assume that Y ∗ = Yn
1
L = (Yt − Yn )2 + α(πt − π̄)2 → min
2
s.t. πt = πte + λ(Yt − Yn )
Then:
1
L = (Yt − Yn )2 + α(πte + λ(Yt − Yn ) − π̄)2 → min
2
∂L
= 2(Yt − Yn ) + αλ(πte + λ(Yt − Yn ) − π̄)2 = 0
∂Yt
(Yt − Yn ) = −αλ(πte + λ(Yt − Yn ) − π̄)
(Yt − Yn ) = −αλ(πt − π̄)
The MP relation defines the level of output that the optimizing CB has to set
depending on the deviation of inflation from the target
Ilya Gulenkov (Higher School of Economics)
Advanced Macroeconomics
Autumn 2023
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Combining MP and IS relations I
IS curve: Yt = multA (A − brt )
MP curve: (Yt − Yn ) = −αλ(πt − π̄)
From the IS relation we have:
Yt = multA (A − brt )
Yn = multA (A − brn )
(Yt − Yn ) = −multA b(rt − rn )
From the MP relation we have:
(Yt − Yn ) = −αλ(πt − π̄)
(Yt − Yn ) = −αλ(πte + λ(Yt − Yn ) − π̄)
(Yt − Yn ) = −αλ2 (Yt − Yn ) − αλ(πte − π̄)
αλ
(Yt − Yn ) = −
(π e − π̄)
1 + αλ2 t
Ilya Gulenkov (Higher School of Economics)
Advanced Macroeconomics
Autumn 2023
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Combining MP and IS relations II
Use the two equations for the output gap:
− multA b(rt − rn ) = −
αλ
(π e − π̄)
1 + αλ2 t
1
e
1 (πt − π̄)
)
multA b(λ + αλ
1
e
rt = rn +
1 (πt − π̄)
multA b(λ + αλ
)
(rt − rn ) =
rt = rn + βπ (πte − π̄)
Combined with the Fisher rule rt = it − πte we get the optimal nominal interest
rate rule:
it = (rn + πte ) + βπ (πte − π̄)
N.B. The rule does not advise the CB to react to the output gap. This result
stems from the assumption of predetermined inflation expectations.
Ilya Gulenkov (Higher School of Economics)
Advanced Macroeconomics
Autumn 2023
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The Taylor Principle
To be able to stabilize the economy, the CB must behave in a way that ensures
an increase in the real interest rate after an increase in inflation expectations,
i.e.
∂it
= 1 + βπ > 1
∂πte
Hence, βπ > 0
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Advanced Macroeconomics
Autumn 2023
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