ECON 4325 Monetary Policy Lecture 7 · Holm Monetary Policy, Lecture 7 32 / 33. Next week I Optimal Monetary Policy under Commitment I Gains from commitment Holm Monetary Policy,

ECON 4325Monetary Policy

Lecture 7

Martin Blomhoff Holm

Outline

1. Implementation of interest rate rules (determinacy).

2. Optimal monetary policy.

3. Rules (commitment) vs. discretion.

4. Optimal monetary policy under discretion.

Holm Monetary Policy, Lecture 7 1 / 33

Part I: Implementation of Interest Rate Rules(Determinacy)


Implementation of Rules I

What do we want? A monetary rule that combined with DIS and NKPCyields a solution consistent with the desired equilibrium outcome.

πt = βEt{πt+1}+ κyt

yt = Et{yt+1} −1

σ(it − Et{πt+1} − rnt )

it = something


Implementation of Rules II

Candidate 1: it = rnt

I Absent shocks, this yields the desired outcome (πt = yt = 0).

I However, the solution is not unique.

I The presence of sunspot shocks can move the economy.


Implementation of Rules III

Candidate 2: it = rnt + φππt + φy yt

I This solution is unique if κ(φπ − 1) + (1− β)φy > 0.

I What does this mean?


Implementation of Rules IV


Implementation of Rules V

I Economic intuition behind κ(φπ − 1) + (1− β)φy > 0?

di = φπdπ + φydy

dπ = βdπ + κdy

What is dr (on blackboard)?

I Interpretation:


What happens if the central bank does not follow theTaylor rule?

I Suppose households permanently increase consumption for no goodreason.

I This results in more production, higher marginal costs, and higherprices.

I The reduction in the real interest rate then justifies the initial increasein consumption.

I The Taylor-rule ensures that this does not happen. There are noself-fulfilling fluctuations.

I Clarida-Gali-Gertler (2000): A change from passive to active monetarypolicy in the early 1980s can explain the ”great moderation.”


Other more subtle points

The model would suggest the following rule

it = rnt + φππt + φy yt ,

but central banks aren’t using that rule. Why not?I rnt is unobservable (and so is yt). To know these, we would need to

haveI exact knowledge of the true model of the economyI exact knowledge of all parameter valuesI exact knowledge of realized values of all shocks

Solution: use optimal simple/robust rules. That is rules based onobservables only, with no need for precise knowledge about model,parameters, or shocks.


Part II: Optimal Monetary Policy


Optimal Monetary Policy I

Introducing the loss function:

Lt =1

2Et

∞∑k=0

βk [λx2t+k + π2t+k ]

I λ is the relative weight on output.

I What does this mean? Think variances.

I Indifference curves are ellipses around zero (draw on blackboard).

I Can be derived from the model, see appendix (second-orderapproximation to household’s welfare):

Lt =1

2Et

∞∑k=0

βk

[(σ +

φ+ α

1− α

)x2t+k +

ε1−θθ (1− βθ) 1−α

1−α+αεπ2t+k

]


Optimal Monetary Policy III

How to choose targets and weights in practice?

Lt =1

2Et

∞∑k=0


I Why target inflation? The uncertainty generated for future plans is acost of inflation volatility.

I Why target the natural level of output? To reach the level of outputthat would prevail in absence of no price rigidities.

I What is the optimal relative weight?I What is the inflation target?

I Should be positive due to measurement errors and downwardnominal wage rigidities.

I Should be close to other similar countries’ target.I Should not be too high to ensure price stability.


Optimal Monetary Policy II

The model (CGG 1999):

xt = −η[it − Etπt+1] + Etxt+1 + gt

πt = κxt + βEtπt+1 + ut

gt = µgt−1 + gt

ut = ρut−1 + ut

Lt =1

2Et

∞∑k=0


I gt is ”demand” shock (similar to discount rate shock).

I ut is ”cost-push” shock. Examples?


Optimal Monetary Policy III

But there are many ways to solve the equation system:

1. Discretion (central bank re-optimizes every period).

2. Commitment (central bank commits to an interest rate path).


Part III: Rules (Commitment) vs. Discretion


Rules vs. Discretion

I Discretion

I Re-optimize every period.I Take expectations as given.I Time-consistent (incentive-compatible)

I ”Rules” (Commitment)

I Commit to a specific reaction pattern.I Take expectation formations as given (try to affect expectations).I Time-inconsistent (not incentive-compatible).


Commitment I

Why commitment? Any gains?

1. Eliminates inflation bias.

2. Can improve current inflation-output trade-off by committing topolicies that affect future expectations.


Commitment II

Two main types:

I Ramsey rule:

I Optimize today. Commit in all future periods.I Exploit initial conditions.

I Timeless perspectiveI As Ramsey, but act as if you committed a long period ago.I Does not exploit the initial conditions.


Commitment III

Is it possible to commit?Is it credible when it is not incentive compatible?


Part IV: Optimal Monetary Policy under Discretion


Optimal Monetary Policy under Discretion

I Time-consistent

I Most in accordance with reality

xt = −η[it − Etπt+1] + Etxt+1 + gt

πt = κxt + βEtπt+1 + ut

gt = µgt−1 + gt

ut = ρut−1 + ut

Lt =1

2Et

∞∑k=0


Take the relative weight, λ, as given.



Two steps:

1. Minimize the loss function with respect to inflation and the outputgap, subject to the NKPC.

2. Conditional on optimal values for inflation and output gap,determined the nominal interest rate from the DIS.



Step I: Minimize the loss function subject to the NKPC.

minxt ,πt

1

2[λx2t + πt ] + Ft

subject to

πt = κxt + ft

where Ft = 12Et

∑∞k=1 β

k [λx2t+k + π2t+k ] and ft = βEtπt+1 + ut .

Why reformulate this way?

I Under discretion, future inflation and output are not affected bytoday’s actions since the central bank re-optimize every period.

I The central bank cannot directly manipulate expectations sinceexpectations are rational.



FOCs (solve on blackboard):

xt = − κ

λ+ κ2ft

πt =λ

λ+ κ2ft

Combining the two FOCs:

xt = −κλπt

Interpretation: leaning against the wind



How to find the reduced form solution?

I Two first order condition + rational expectations.

I Two ways to find the optimal solution:

1. Method of undetermined coefficients2. Solve FOC wrt π forward.



Solve the FOC with method of undetermined coefficient to obtain (onblackboard):

πt =λ

κ2 + λ(1− βρ)ut

and insert this solution into the leaning against the wind condition toobtain

xt = −κλπt = − κ

κ2 + λ(1− βρ)ut

and insert this into the DIS-equation (xt = −η[it − Etπt+1] + Etxt+1 + gt)to find the interest rate

it =κ(1− ρ) + ρλη

η(κ2 + λ(1− βρ))ut +

1

ηgt




η(κ2 + λ(1− βρ))ut +

1

ηgt

I How does the central bank react to a demand shock?

I How does the central bank react to a cost-push shock?

Interpretations:


Key Result I: Trade-off

Key Result I: Trade-off between output and inflation variability whencost-push shocks exist.

Efficient policy frontier:

Var(xt) = Et(x2t )− [Et(xt)]2

=

(κ

κ2 + λ(1− βρ)

)2

Et(u2t )−

[κ

κ2 + λ(1− βρ)Etut

]2=

(κ

κ2 + λ(1− βρ)

)2

σ2u

⇒ σx =κ

κ2 + λ(1− βρ)σu

and for inflation:

σπ =λ

κ2 + λ(1− βρ)σu


Key Result I: Trade-off

I What happens when λ→ 0?

I Will there be a trade-off in the absence of cost-push shocks?


Key Result II: Inflation Targeting

Key Result II: Optimal monetary policy incorporates inflation targeting.

Optimal monetary policy aims for convergence of inflation to its targetover time when cost-push shocks exist and the relative weight on outputstabilization is non-zero.

I Leaning against the wind.

I Cost-benefit: λ determines the trade-off.

Extreme inflation targeting is only optimal if either

I When cost-push shocks are absent (σu = 0).

I When the weight on output stabilization is zero (λ = 0).


Key Result III: Taylor condition

Key Result III: Optimal policy requires that the nominal interest rateshould be adjusted by more than one-for-one to changes in inflationexpectations.


η(κ2 + λ(1− βρ))ut +

1

ηgt =

(1 +

κ(1− ρ)

ηρλ

)Etπt+1 +

1

ηgt


Key Result IV: Demand Shocks

Key Result IV: Optimal policy requires that the nominal interest rateshould be adjusted to perfectly offset demand shocks and not react toshocks to potential output.

I No trade-off with demand shocks.

I No trade-off from productivity shocks either.I Permanent rise in output raises potential output and actual output by

the same amount, i.e. no change in the output gap.

I The central bank therefore do not adjust the interest rate when thereare productivity shocks.

Challenge for monetary policy: Distinguish between the shocks.


Next week

I Optimal Monetary Policy under Commitment

I Gains from commitment


Documents

ECON 4325 Monetary Policy Lecture 7 · Holm Monetary Policy, Lecture 7 32 / 33. Next week I Optimal Monetary Policy under Commitment I Gains from commitment Holm Monetary Policy,