advmacro Takeki Sunakawa NKMP2

Monetary Policy Tradeoffs:

Discretion vs. Commitment

Takeki Sunakawa

Advanced Macroeconomics at Tohoku University

Introduction

The efficient allocation is obtained under fully flexible prices. The optimal policy fully stabilizes the price level.

In practice, central banks face short-run tradeoffs: inflation vs. real variables such as output and employment.

We need a monetary policy design in environment in which the central bank faces a nontrivial tradeoff.

The case of an efficient steady state

Consider a situation in which the flexible price equilibrium allocation is inefficient. The natural level of output y_tⁿ deviates from its efficient counterpart y^e_t in the short run. On the other hand, yⁿ= y^eholds in the long run.

Some real imperfections generates a time-varying gap ut≡ y_tⁿ− y^e_t even in the absence of price rigidities.

The CB’s problem

The central bank (CB hereafter) will minimize the welfare loss function:

E0

X∞ t=0

β^t π²t+ ϑx²t

,

where xt≡ yt− yt^eis the welfare-relevant output gap with yt^edenoting the efficient level of output. ϑ = κ/ǫ is the weight of output gap fluctuations. Minimization is subject to

πt= βEtπt+1+ κxt+ ut, where ut≡ κ(y^et− ytⁿ).

[Note that ˜xt= yt− yⁿ_t = (yt− y_t^e)

| {z }

xt

+ (y^e_t− y_tⁿ)

| {z }

ut

.]

Cost-push shock

ut follows the exogenous AR(1) process

ut= ρuut−1+ ε^ut,

where ρu∈ [0, 1) and ε^ut is white noise with variance σ²u. ut generates a tradeoff between stabilizing πt and stabilizing xt.

Optimal Discretionary Policy

Each period the CB minimizes the period losses πt²+ ϑx²t, subject to the constraint

πt= κxt+ νt,

where νt≡ βEtπt+1+ utis taken as given by the central bank, as there are no endogenous state variables.

Optimal Discretionary Policy, cont’d

The optimality condition is given by xt= −^κ

ϑ^πt, for t = 0, 1, 2, ....

Substituting it into the NKPC and after some manipulation (e.g., undetermined coefficient method), we have

πt = ^ϑ

κ²+ ϑ(1 − βρ)^ut,

πt = − ^κ

κ²+ ϑ(1 − βρ)^ut.

Optimal Commitment Policy

Now we assume that the CB is able to commit to future policies. The CB will choose a state-contingent sequence {xt, πt}^∞_t=0so as to minimize

E0

X∞ t=0

β^t π²t+ ϑx²t

,

subject to

πt= βEtπt+1+ κxt+ ut, and utfollows the AR(1) process:

ut= ρuut−1+ ε^ut.

Optimal Commitment Policy, cont’d

It is useful to set up the Lagrangian as

L = E0

X∞ t=0

β^{t 1} 2 ^π

2 t ^{+ ϑx}

2 t

+ ξt(πt− βEtπt+1− κxt)

 ,

where {ξt} is a sequence of Lagrange multipliers. The FONCs are

ϑxt− κξt= 0, πt+ ξt− ξt−1= 0,

for t = 0, 1, 2, ...,given ξ−1= 0, because there is no commitment in period 0.

Optimal Commitment Policy, cont’d

Combining the two, we have

x0= −^κ ϑ^π0, and

xt− xt−1= −^κ ϑ^πt for t = 1, 2, 3, ....

Solving for the policy function under commitment

We assume: xt= axut+ bxxt−1and πt= aπut+ bπxt−1with the initial condition x−1= 0.

Substitute them into the NKPC and the tradeoff equation, aπut+ bπxt−1 = βEt(aπut+1+ bπxt) + κxt+ ut,

= (1 + βρuaπ)ut+ (βbπ+ κ)(axut+ bxxt−1),

= (1 + (βbπ+ κ)ax+ βρuaπ)ut+ (βbπ+ κ)bxxt−1.

aπut+ bπxt−1 = −(ϑ/κ) (xt− xt−1) ,

= −(ϑ/κ) (axut+ (bx− 1)xt−1) .

Solving for the policy function under commitment, cont’d

Then we have

aπ = −(ϑ/κ)ax

bπ = (ϑ/κ)(1 − bx)

aπ = 1 + (βbπ+ κ)ax+ βρuaπ, bπ = (βbπ+ κ)bx

These can be solved for

ax= −(κ/ϑ)/[β(δ⁺− ρu)], aπ= 1/[β(δ⁺− ρu)], bx= δ⁻∈ (0, 1), bπ= (ϑ/κ)(1 − δ⁻). where δ^± =^1 ±^p1 − 4βγ^2/ (2βγ) is the solution of a quadratic equation βγδ²− δ + γ = 0 where γ = (1 + β + κ²/ϑ)⁻¹.

Discretion vs. Commitment: ρ

= 0.0 (when α = 0)

0 5 10

-2 -1.5 -1 -0.5

0 ^{Output gap}

0 5 10

-0.2 -0.1 0 0.1

0.2 ^Inflation

0.1 0.15

0.2 Price level

0.5 1

1.5 ^u

Discretion vs. Commitment: ρ

= 0.8 (when α = 0)

0 5 10

-2 -1.5 -1 -0.5

0 ^{Output gap}

0 5 10

-0.1 0 0.1 0.2

0.3 ^Inflation

0.2 0.4 0.6 0.8

1 Price level

0.5 1

1.5 ^u

The case of a distorted steady state

Consider the case in which there is a permanent gap x ≡ yⁿ− y^e. Specifically,

−^Un Uc

= (1 − Φ)M P N,

where Φ ≥ 0 measures the wedge between the marginal product of labor and marginal rate of substitution.

For example, monopolistic competition and associated markup is a source of the distortion, Φ ≡ 1 − [(1 − τ )M]⁻¹≥ 0.

Second-order approximation to the household utility

Note that −Un/Uc= (1 − Φ)M P N = (1 − Φ)C/N implies (1 − Φ)UcC = −UnN . Then we have

Ut− U UcC ^≃

 ˆ

yt(1 + zt) +^{1 − σ} 2 ^yˆ

2 t



−(1 − Φ)

 ˆ

yt(1 + zt) + dt+^{1 + ϕ}

2 ^{(ˆyt− at)}

2



+ t.i.p.,

= Φ

 ˆ

yt(1 + zt) + dt+^{1 + ϕ}

2 ^{(ˆyt− at)}

2



−



dt+^{σ + ϕ} 2 ^yˆ

2

t + (σ + ϕ)ˆytat



+ t.i.p.,

= Φˆyt−



dt+^{σ + ϕ} 2 ^yˆ

2

t + (σ + ϕ)ˆytat



+ t.i.p.,

Under the small distortion assumption, the product of Φ with second-order terms is negligible. Also, Φˆytcan be considered as a second-order term.

The CB’s problem: The case of small SS distortions

Under the small distortion assumption, the welfare loss function is given by

E0

X∞ t=0

β^{t 1} 2 ^π

2 t + ϑˆx²t

− Λˆxt

 ,

where Λ ≡ Φλ/ǫ > 0 and ˆxt≡ xt− x with x ≡ yⁿ− y^e. Similarly, the NKPC can be written as

πt= βEtπt+1+ κˆxt+ ut.

Optimal Discretionary Policy

Each period the CB minimizes the period losses 1

2 ^π

2 t ^{+ ϑˆx}

2 t

− Λˆxt,

subject to the constraint

πt= κˆxt+ νt,

where νt≡ βEtπt+1+ utis taken as given by the central bank.

Optimal Discretionary Policy, cont’d

The optimality condition is given by ˆ xt=^Λ

ϑ ⁻ κ ϑ^πt, for t = 0, 1, 2, ....

Substituting it into the NKPC and we have

πt = ^Λκ

κ²+ ϑ(1 − β) ⁺

ϑ

κ²+ ϑ(1 − βρ)^ut, ˆ

xt = ^{Λ(1 − β)} κ²+ ϑ(1 − β) ⁻

κ

κ²+ ϑ(1 − βρ)^ut. The constant terms are known asthe inflation bias.

Optimal Commitment Policy

The Lagrangian is set up as

L = E0

X∞ t=0

β^{t 1} 2 ^π

2 t^{+ ϑˆx}

2 t

− Λˆxt+ ξt(πt− βEtπt+1− κˆxt)

 ,

where {ξt} is a sequence of Lagrange multipliers. The FONCs are

ϑxt− κξt− Λ = 0, πt+ ξt− ξt−1= 0, for t = 0, 1, 2, ... and ξ−1= 0.

Optimal Commitment Policy, cont’d

Combining the two, we have

ˆ x0= −^κ

ϑ^π0+ Λ ϑ^, and

ˆ

xt− ˆxt−1= −^κ ϑ^πt for t = 1, 2, 3, ....

Note that the equilibrium conditions has the same form in the case of the efficient steady state except for t = 0.

ˆ

x−1 = Λ/ϑ is given as the initial condition.

Initial dynamics

0 5 10

0 0.5 1 1.5

2 ^{Output gap}

0 5 10

0 0.05 0.1 0.15 0.2

0.25 ^Inflation

1 2

3 Price level

-0.5 0 0.5

1 ^u

The case of large SS distortions

When the steady state distortions are large, the naive LQ approximation is no longer correct.

There are two (three?) alternative approaches to overcome the problem.

1 Use a second-order approximation of the structural equations to replace the linear terms in the welfare function to obtain the correct LQ approximation (Benigno and Woodford, 2005).

2 Solve the original nonlinear Ramsey problem for the equilibrium conditions. Then log-linearize the resulting equilibrium conditions (Khan, King and Wolman, 2003).

The zero lower bound on nominal interest rates

Money has no nominal payoffs but otherwise is identical to short-term nominal debt. This fact may impose the zero lower bound (ZLB) on the nominal return of such debt:

it≥ 0. Then the CB minimizes

E0

X∞ t=0

β^t π²t+ ϑx²t

,

subject to

πt = βEtπt+1+ κxt,

xt ≤ xt+1+ σ⁻¹(πt+1+ rtⁿ) .

The last equation is anoccasionally binding constraint. The equality holds when it= 0.

The natural rate and the ZLB

We assume the natural rate rⁿ_t follows an exogenous deterministic path: In the steady state, r_tⁿ= ρ. Itunexpectedlydrops to and remains at

rⁿt = −ǫ < 0 for t = 0, 1, ..., tZ. From period tZ+ 1 onward, it reverts to rⁿt = ρ again.

We assumethe perfect foresight(get rid of expectational operators

hereafter). Agents know the subsequent path of the natural rate in period 0.

Whenever rⁿt < 0, the efficient allocation, implied by it= rⁿt, is no longer attainable.

Optimal Discretionary Policy

Each period the CB minimizes the period losses πt²+ ϑx²t, subject to the constraint

πt = κxt+ ν0,t, xt ≤ ν1,t,

where ν0,t≡ βEtπt+1 and ν1,t≡ xt+1+ σ⁻¹(πt+1+ rtⁿ) are taken as given.

Optimal Discretionary Policy, cont’d

The Lagrangian is given by L = ¹

2 ^π

2 t+ ϑx²t

+ ξ1,t(πt− κxt− ν1,t) + ξ2,t(xt− ν2,t) .

The FONCs are

xt+ ξ1,t= 0, ϑxt− κξ1,t+ ξ2,t= 0, and the slackness conditions

ξ2,t≥ 0, it≥ 0, ξ2,tit= 0. for t = 0, 1, 2, ....

The solution

From t = tZ+ 1 onward, it= ρ > 0, ξ2,t= 0, and xt= πt= 0 hold. For t = 0, 1, ..., tZ, it= 0, ξ2,t> 0, and ϑxt= −κπt− ξ2,thold. Then we can solve the two equations backward

πt = βπt+1+ κxt,

xt = xt+1+ σ⁻¹(πt+1+ r_tⁿ) , from t = tZ, tZ− 1, ..., 0, given xt_Z+1 = πt_Z+1 = 0.

Optimal Commitment Policy

The CB will choose a state-contingent sequence {xt, πt}^∞_t=0so as to minimize

E0

X∞ t=0

β^t π²_t+ ϑx²_t
,

subject to

πt = βπt+1+ κxt,

xt ≤ xt+1+ σ⁻¹(πt+1+ rtⁿ) .

Optimal Commitment Policy, cont’d

It is useful to set up the Lagrangian as

L = E0

X∞ t=0

β^{t 1} 2 ^π

2 t ^{+ ϑx}

2 t

+ ξ1,t(πt− βπt+1− κxt)

+ξ2,t xt− xt+1− σ⁻¹(πt+1+ rⁿt)
^. The FONCs are

πt+ ξ1,t− ξ1,t−1− (βσ)⁻¹ξ2,t−1 = 0, ϑxt− κξ1,t+ ξ2,t− β⁻¹ξ2,t−1 = 0, and the slackness conditions

ξ2,t≥ 0, it≥ 0, ξ2,tit= 0.

for t = 0, 1, 2, ..., given the initial conditions ξ1,−1= ξ2,−1= 0.

A conjectured solution

The solution is conjectured and verified as follows;

1 From t = 0 to tC ≥ tZ , it= 0 and ξ2,t> 0.

2 _{For t = tC+ 1, it= 0, ξ2,t= 0 and ξ2,t> 0.}

3 _{From t = tC}+ 2 and onward, it> 0 and ξ2,t= ξ2,t−1 = 0,

t = t

+ 2 and onward

Note that ξ2,t= ξ2,t−1 = 0, then

πt+ ξ1,t− ξ1,t−1= 0, ϑxt− κξ1,t= 0, πt= βπt+1+ κxt, hold, given the initial condition ξ1,tC+1.

These equations can be solved for

xtC+2+k = −^κδ

k+1

ϑ ^ξ1,tC+1, πtC+2+k = (1 − δ)δ^kξ1,tC+1. for k = 0, 1, 2, .... [check by yourself]

t = t

+ 1

Note that itC+1> 0 and ξ2,tC+1= 0, then

πtC+1+ ξ1,tC+1− ξ1,tC − (βσ)⁻¹ξ2,tC = 0, ϑxtC+1− κξ1,tC+1− β⁻¹ξ2,tC = 0,

πtC+1= β (1 − δ)ξ1,tC+1

| {z }

=π_{tC +2}

+κxtC+1,

hold.

By substituting out ξ1,tC+1= [β(1 − δ)]⁻¹(πtC+1− κxtC+1), we have [1 + β(1 − δ)]πtC+1− κxtC+1− β(1 − δ)ξ1,tC− [(1 − δ)/σ]ξ2,tC = 0, [β(1 − δ)ϑ + κ²]xtC+1− κπtC+1− (1 − δ)ξ2,tC = 0.

t = 0, 1, ..., t

^{and so on}

Note that it= 0, then

πt = βπt+1+ κxt,

xt = xt+1+ σ⁻¹(πt+1+ rtⁿ) , hold, where

rⁿt =

(−ρ, when t = 0, 1, ..., tZ, ρ, when t = tZ+ 1, ..., tC. In addition, the original FONCs

πt+ ξ1,t− ξ1,t−1− (βσ)⁻¹ξ2,t−1 = 0, ϑxt− κξ1,t+ ξ2,t− β⁻¹ξ2,t−1= 0,

hold for t = 0, ..., tC+ 1, given the initial conditions ξ1,−1= ξ2,−1= 0.

The algorithm for the solution

The algorithm is as follows:

1 _{Fix tC≥ tZ}. Solve 4 × (tC+ 2) equations for 4 × (tC+ 2) variables {xt, πt, ξ1,tξ2,t}^tt=0^C+1.

2 _{Obtain xt}

C+2+k = −^κδk+1_ϑ ξ1,tC+1 and πtC+2+k= (1 − δ)δ^kξ1,tC+1 for k = 0, 1, ..., given ξ1,tC+1 at hand.

3 _{Check it= r}ⁿ

t + πt+1+ σ(xt+1− xt) = 0 for t = 0, 1, ...tC and it> 0 t = tC+ 1, .... If not, increase tC and do 1-3 again.

The relevant equations

 xt

πt



=

 1 _σ¹ κ β +^κ_σ



| {z }

A

 xt+1

πt+1

 +

 1 σκ σ

 rⁿ_t,

for t = 0, 1, ..., tC,

 xtC+1

πtC+1



=

 −κ 1 + β(1 − δ)

β(1 − δ) +^κ_ϑ² −^κ_ϑ

⁻¹

β(1 − δ) ^1−δ_σ 0 ^1−δ_ϑ



| {z }

M

×

 ξ1,tC

ξ2,tC

 ,

 ξ1,t

ξ2,t



=

 1 (βσ)⁻¹ κ β⁻¹(1 + κσ⁻¹)



| {z }

H

 ξ1,t−1

ξ2,t−1



−

 0 1 ϑ κ



| {z }

J

 xt

πt

 ,

The relevant equations, cont’d

These equations can be stacked into



















I2 −A

J _I2

I2 −A

−H ^J I2

. .. . ..

−M I2

−H ^J I2









































 x0

π0

ξ1,0

ξ2,0

x1

π1

ξ1,1

ξ2,1

... xtC+1

πtC+1

ξ1,tC+1

























=



























rⁿ₀ κrσⁿ₀

σ

0 0

rⁿ₁ κrσⁿ₁

σ

0 0 ... 0 0 0

























 .

Discretion vs. Commitment with the ZLB (when α = 0)

0 5 10

-15 -10 -5 0

5 ^{Output gap}

0 5 10

-60 -40 -20 0

20 ^Inflation

0 2 4

6 Nominal rate

0

5 Natural rate

Caveats

We have assumed the perfect foresight.

Uncertaintyabout the natural rate matters. We need to solve the model with stochastic settings (e.g., Adam and Billi, 2006; 2007, Nakov, 2008).