Technical Appendix to Optimal Monetary P

A

Analytical Details

A.1

Households

Cost Minimization

Households decide the composition of the consumption basket

to minimize expenditures
min
{Citk }i
s.t.



0

1

Citk



1
0

Pit Citk di

− θCit−1

 η−1
η

η
 η−1
k
di

≥ Xt

The demand for individual goods i is
Citk

=



Pit
Pt

−η

Xtk + θCit−1 ,

where Pt is the overall price level, expressed as an aggregate of the good i prices, Pt =

 1
1−η

1 1−η
P
di
.
0 it
Utility Maximization

The solution to the utility maximization problem is ob-

tained by solving the Lagrangian function,
∞






k
L = E0 β t u Xtk , Ntk − λkt Pt Xtk + Pt ϑt + Qt,t+1 Dt+1
− Wt Ntk (1 − τ ) − Dtk − Φt − Tt .

t=0

32

In the budget constraint, we have re-expressed the total spending on the consumption
1
basket, 0 Pit Citk di, in terms of quantities that affect the household’s utility,


0

1

Pit Citk di = Pt Xtk + Pt ϑt ,

where under deep habits ϑt is given as ϑt ≡ θ

 1  Pit 
0

Pt

Cit−1 di, while under superficial

habits it takes the simpler form, ϑt ≡ θCt−1 . Households take ϑt as given when maximizing utility.
The first order conditions are then,







Xtk :


Ntk :

Dtk :

where Rt =

uX (t) = λkt Pt
t
−uN (t) = uX (t) W
Pt (1 − τ )

1 = βEt



1
Et [Qt,t+1 ]

uX (t+1) Pt
uX (t) Pt+1

Rt

is the one-period gross return on nominal riskless bonds.

With utility given by u (X, N ) =
uX (·) = X −σ

X 1−σ
1−σ

−

N 1+υ
1+υ ,

and

33

the first derivatives are

uN (·) = −N υ .

A.2

Firms

The cost minimization involves the choice of labor input Nit subject to the available
production technology
minWt Nit
Nit

s.t. At Nit = Yit
The minimization problem implies a labor demand, Nit =

Yit
At ,

and a nominal marginal

t
cost which is the same across all brand producing firms, M Ct = (1 − κ) W
At . Profits are

defined as:

Φit

2
Pit
− 1 Pt Yt
πPit−1
2

Pit
ϕ
= (Pit − M Ct ) Yit −
− 1 Pt Yt
2 πPit−1
ϕ

≡ Pit Yit − Wt Nit −
2



where the last term represents the nominal costs of adjusting prices, as in Rotemberg
(1982), and π is the steady state inflation.
Each firm then chooses processes for Pit and Yit to maximize the present discounted
value of profits, under the restriction that all demand be satisfied at the chosen price
(Cit = Yit ):
max Et

{Pit , Yit }

∞


Qt,t+s Φit+s = Et

s=0

s.t.Yit+s
Qt,t+s



∞


Qt,t+s

s=0



ϕ
(Pit+s − M Ct+s ) Yit+s −
2


Pit+s −η
=
Xt+s + θYit+s−1
Pt+s

−σ
Pt
s Xt+s
= β
Xt
Pt+s




2
Pit+s
− 1 Pt+s Yt+s
πPit+s−1

The first order conditions are:
vit = (Pit − M Ct ) + θEt [Qt,t+1 vit+1 ]
and
  
 




Pt Yt
Pit+1
Pit
Pit+1
Pit −η−1 Xt
−1
− ϕEt Qt,t+1
−1
+ ϕ
Yit = vit η
Pt+1 Yt+1
Pt
Pt
πPit−1
πPit−1
πPit
π (Pit )2
where vit is the Lagrange multiplier on the dynamic demand constraint and represents
the shadow price of producing good i.

34

B

Equilibrium Conditions

B.1

Aggregation and Symmetry

Aggregate output:

In this setup, all firms and all households are symmetric. This

implies that aggregate output is given by
Yt = At Nt
Aggregate resource constraint: Aggregate profits are
Φt = Pt Yt − Wt Nt −

2
ϕ  πt
− 1 Pt Yt
2 π

and the household budget constraint becomes in equilibrium (note: Pt Xt + Pt ϑt reduces
to Pt Ct )
Pt Ct = Wt Nt (1 − τ ) + Φt + Tt
Combine the household budget constraint with the government budget constraint (τ Wt Nt = Tt )
and the definition of profits to obtain the aggregate resource constraint
Ct +

B.2

2
ϕ  πt
− 1 Yt = Yt
2 π

System of Non-linear Equations
Xt = Ct − θCt−1
Wt
≡ wt (1 − τ )
Pt
 −σ

Rt π −1
Xt−σ = βEt Xt+1
t+1


−σ 
π
π
π
X
π
t+1
t+1
t
t+1
t
−1
Yt − ϕβEt
−1
Yt+1
Yt = ηω t Xt + ϕ
π
π
Xt
π
π





Xt+1 −σ
1
ω t+1
+ θβEt
ωt = 1 −
μt
Xt
Ntυ Xtσ =

Yt = At Nt
2
ϕ  πt
Y t = Ct +
− 1 Yt
2 π
wt
mct =
At
μt =

1
mct

ln At = ρ ln At−1 + ǫt

35

(32)
(33)
(34)
(35)

(36)
(37)
(38)
(39)
(40)
(41)

B.3

The Deterministic Steady State

The non-stochastic long-run equilibrium is characterized by constant real variables and
nominal variables growing at a constant rate. The equilibrium conditions (32) - (41)
reduce to:
X = (1 − θ) C

(42)

N υ X σ = w(1 − τ )


1 = β Rπ −1 = βr

(43)

Y = ηωX

(45)

μ = [1 − (1 − θβ) ω]−1

(46)

Y = AN

(47)

Y =C

(48)

w
A
1
μ=
mc

mc =

(44)

(49)

A=1
Table 1 contains the imposed calibration restrictions. We assume values for the real
interest rate, the Frisch labor supply elasticity, steady state inflation, and the parameters
σ, η, ϕ, and θ. The discount factor β matches the assumed real rate of interest, β = r −1 ,
while the nominal interest rate is R = rπ. Given the specification of the utility function,
υ is the inverse of the Frisch labor supply elasticity, ǫN w = υ1 .
With no price adjustment costs in the deterministic steady state, C = Y. Then, using
equations (42) and (45), the steady state value of the shadow price ω is
ω = [η (1 − θ)]−1 ,
while the markup μ is given by equation (46) and the marginal cost is its inverse, mc =
μ−1 .
To determine the steady state value of labor, we substitute for X in terms of Y in (43)
and then, using the aggregate production function, we obtain the following expression,
N σ+υ [(1 − θ) A]σ = w(1 − τ ),

(50)

which can be solved for N . Note that this expression depends on the real wage w, which
can be obtained from equation (49). However, in order to assess the level of taxation
needed to make the long-run equilibrium efficient, we substitute for real wages using the

36

steady-state condition for marginal costs, mc = w/A = 1/μ
N σ+υ [(1 − θ) A]σ =

A
(1 − τ ),
μ

(51)

In order for this condition to match the social planner’s allocation (65) it must be the
case that τ = 1 − μ (1 − θβ). See Appendix E for the social planner’s problem. Finally,
equations (47) and (42) can be solved for aggregate output Y (or consumption C) and
habit-adjusted consumption X.

B.4

System of Log-linear Equations

Log-linearizing the equilibrium conditions (32) - (41) around the efficient deterministic
steady state gives the following set of equations:


t = (1 − θ)−1 C
t − θC
t−1
X

t + υ N
t = w
σX
t


t = Et X
t+1 − 1 R
t − Et π
X
t+1
σ

t + ϕ (
Yt = ω
t + X
π t − βEt π
t+1 )


1
t − Et X
t+1
ω
t =
μ
t + θβEt ω
 t+1 + θβσ X
μω
t + N
t
Yt = A
t
Yt = C

t
mc
t = w
t − A
μ
t = −mc
t

t = ρA
t−1 + εt
A

The New Keynesian Phillips Curve is given by the pricing equation (52)
π
t = βEt π
t+1 +


1 
t
Yt − ω
t − X
ϕ

where the evolution of the shadow value ω
 t is given by equation (53).

37

(52)
(53)

C

The Case of Superficial External Habits

C.1

Households

Habits are “superficial” when they are formed at the level of the aggregate consumption
good. Households derive utility from the habit-adjusted composite good Xtk ,
Xtk = Ctk − θCt−1 ,
where household k’s consumption, Ctk , is an aggregate of a continuum of final goods,
indexed by i ∈ [0, 1] ,
Ctk

=



0

1

Citk

 η−1
η

η
 η−1
di
,

with η > 1 the elasticity of substitution between them and Ct−1 ≡
sectional average of consumption.
Cost Minimization

1
0

k dk the crossCt−1

Households decide the composition of the consumption basket

to minimize expenditures


1

Pit Citk di
min
k
{Cit }i 0
η
 1   η−1  η−1
k
η
k
di
≥ Ct .
Cit
s.t.
0

The demand for individual goods i is
Citk

where Pt ≡


1
0

=



Pit
Pt

−η

Ctk ,

 1
1−η
Pit1−η di
is the consumer price index. The overall demand for good i

is obtained by aggregating across all households
Cit =



1
0

Citk dk =



Pit
Pt

−η

Ct .

(54)

Unlike in the case of deep habits, this demand is not dynamic.

C.2

Firms

The firms’ cost minimization problem is unchanged, while the profit maximization is still
dynamic (due to the nature of price inertia) but subject to the static demand (54). The

38

price is set optimally to satisfy the following relationship:







π
π
π
Xt+1 −σ  π t+1
1
t
t+1
t
−1
Yt − ϕβEt
−1
Yt+1 .
Yt = ϕ
1−η 1−
μt
π
π
Xt
π
π
(55)

C.3

Equilibrium

In this setup, we obtain the familiar looking New Keynesian Phillips Curve,

to which we add the IS curve,

π
t = βEt π
t+1 +

η−1
mc
t
ϕ

t + 1 Et π
t = Et X
t+1 − 1 R
t+1 ,
X
σ
σ

(56)

(57)

and two equations defining the habit-adjusted consumption and the real marginal cost,
t =
X


1 
Yt − θYt−1
1−θ

t + υ Yt − (1 + υ) A
t .
mc
 t = σX

39

(58)
(59)

D

The Case of Internal Habits

D.1

Households

Habits are internal and superficial when they are formed at the level of the aggregate
consumption bundle but households endogenize the effects of their current consumption
choices on future utility. Each household k derives utility from a habit-adjusted composite
good Xtk ,
k
Xtk = Ctk − θCt−1

where Ctk is the time-t aggregate of a continuum of goods, indexed by i ∈ [0, 1] ,
Ctk

=



0

1

Citk

 η−1
η

η
 η−1
di

k
is the previous period
with η > 0 the elasticity of substitution between them, and Ct−1

consumption of household k. Since households are symmetric and, in this case, we also
do not need to distinguish between individual and aggregate variables, in what follows we
drop the k superscript.
Cost Minimization

The households’ cost minimization problem yields a typical

static demand for each good i,
Cit =
where Pt ≡


1
0

 1
1−η
Pit1−η di
.



Pit
Pt

−η

Ct

Utility maximization

The Lagrangian function is





∞

L = E0 β t
t=0

where Rt =

N 1+υ
(Ct − θCt−1 )1−σ
− t
1−σ
1+υ

1
Et [Qt,t+1 ]



− λt (Pt Ct + Et Qt,t+1 Dt+1 − Wt Nt (1 − τ ) − Dt − Φt − Tt )

is the one-period gross return on nominal riskless bonds. Anticipat-

ing that current consumption decisions affect the stock of habits entering into the future,
the consumption Euler equation is given by,
Xt−σ

−

−σ
θβEt Xt+1

= βEt



 Rt
 −σ
−σ
Xt+1 − θβEt+1 Xt+2
π t+1

and the labor supply condition by,
(Nt )υ
 −σ

−σ = wt (1 − τ )
Xt − θβEt Xt+1

40



In log-linear form, the first order condition for labor and the Euler equation become

and

D.2

t = w
υN
t −


σ 
t+1
Xt − θβEt X
1 − θβ

(60)





t − θβEt X
t+1 = Et X
t+1 − θβEt+1 X
t+2 − 1 − θβ R
t − Et π
X
t+1
σ

(61)

Firms

The firms’ behavior is unchanged, except that the stochastic discount factor in their profit
maximization problem is given by,
Qt,t+s = β s



λt+s
λt



= βs



−σ
−σ
Xt+s
− θβEt+s Xt+s+1
−σ
Xt−σ − θβEt Xt+1



Pt
Pt+s

and the firms’ FOC for price is then
[1 − η (1 − mct )] Yt = ϕ

π
t
−1
Yt −ϕβEt
π
π

π

t



−σ
−σ
Xt+1
− θβEt+1 Xt+2
−σ
Xt−σ − θβEt Xt+1



π

t+1

π

π
t+1
−1
Yt+1
π

In a deterministic steady state, this relationship still reduces to 1 = η (1 − mc) , and
the log-linearized equation, which is essentially the NKPC, is the same as under external
superficial habits.
The marginal cost relationship is slightly modified due to the different marginal rate
of substitution between consumption and labor:
t
mc
t = w
 −A
t


σ 


t
=
Xt − θβEt Xt+1 + υ Nt − A
1 − θβ

σ 
t
t+1 + υ Yt − (1 + υ) A
Xt − θβEt X
=
1 − θβ

(62)

Hence, the system of relevant equations includes the IS curve (61), the NKPC (56), the
marginal cost relationship (62), and the usual definition of the habit-adjusted consumption
(58).
In gap form
Ytg

In the case of internal habits, it is easy to write these relationships in ‘gap’ form. Let
g ≡ X
t − X
t∗ denote the relevant ‘gap’ variables. Using the equations
≡ Yt − Yt∗ and X
t

describing the social planner’s allocation below, the marginal cost equation can be written

41



as
mc
t =

=

=

=


σ 
t
t+1 + υ Yt − (1 + υ) A
Xt − θβEt X
1 − θβ




σ 
∗
t∗ − θβEt X
t+1 + υ Yt − σ
t+1
− υ Yt∗
Xt − θβEt X
X
1 − θβ
1 − θβ






σ  
∗
t+1 − X
t+1
t∗ − θβEt X
+ υ Yt − Yt∗
Xt − X
1 − θβ

σ  g
g
g
Xt − θβEt X
t+1 + υ Yt
1 − θβ

and the NKPC is then,




η−1
σ  g
g
g


π
t = βEt π
t+1 +
Xt − θβEt X
t+1 + υ Yt
ϕ
1 − θβ

(63)

The IS curve can also be written using gap variables and additional terms involving
∗
∗
only the
social planner’s
allocation. From the social planner’s problem, Xt − θβEt Xt+1 =
1−θβ
t − υ Yt∗ , which then allows us to write the IS curve as (add and subtract
(1 + υ) A
σ

the time t and (t + 1) expressions from the LHS and RHS of the IS equation):

 g − θβEt X
g =
X
t
t+1

D.3



⎧
g
g


⎪
E
X
−
θβE
X
t
t+1
⎪
t+1
t+2 −
⎨

1−θβ
σ


 ⎫
t − Et π
R
t+1 ⎪
⎪
⎬




⎪
⎪
⎪
⎪
⎩ + 1−θβ (1 + υ) (ρ − 1) A
⎭
t − υ Et Y ∗ − Y ∗
t
t+1
σ

Optimal Policy: internal habits

In the case of internal (superficial) habits, we can show analytically that in response to
technology shocks the Ramsey planner can set the nominal interest rate so as to match
the social planner’s allocation,
without
creating inflation. Using the link between gap


g
g
g
1


 =
variables, X
t
1−θ Yt − θ Yt−1 , we first express all relevant equations in terms of

inflation and output gap. The welfare function Γ0 is

 

∞
2
 2

1
g
g
g
2
t
Γ0 = − ΩE0
β δ Yt − θYt−1 + υ Yt
+ ϕ
π t + tip + O[2]
2
t=0

and the New Keynesian Phillips curve (63) is
π
t = βEt π
t+1 −



η−1
g
g
δ θβEt Yt+1
− ζ Ytg + θYt−1
ϕ

where Ω, δ, and ζ are defined as: Ω ≡ N

1+υ

,δ≡

42

σ
(1−θβ)(1−θ) ,

(64)



and ζ ≡ 1 + θ 2 β + υδ .

With the IS curve not binding, the Lagrangian is
⎧
⎪
⎨

 

2
g
g
g
2



πt
δ Yt − θYt−1 + υ Yt + ϕ
βt
L = E0




⎪
⎩ −γ t π
g
g
g
t=0
t − β
π t+1 + η−1
ϕ δ θβEt Yt+1 − ζ Yt + θ Yt−1
∞


− 12 Ω

and the first order conditions for the output gap and inflation are:

⎫
⎪
⎬
⎪
⎭







 
η−1
η−1
η−1
g
g
g
g




γ = θβEt Yt+1 −
γ
γ
+ θ Yt−1 −
: ζ Yt −
Yt
ϕΩ t
ϕΩ t+1
ϕΩ t−1
(
πt ) : π
t = −


1 
γ t − γ t−1
ϕΩ

with the additional restriction that, under full commitment, the central bank ignores
past commitments in the first period and sets all pre-existing conditions to zero, Y g =
−1

γ −1 = 0. By varying interest rates to eliminate the output gap, the value of the Lagrange

multiplier associated with the NKPC is zero, γ t = 0, and the policy maker can achieve the
flexible-price allocation which is desirable since there are no frictions other than nominal
inertia in this economy featuring internal habits.

43

E

The Social Planner’s Problem

The subsidy level that ensures an efficient long-run equilibrium is obtained by comparing
the steady state solution of the social planner’s problem with the steady state obtained in
the decentralized equilibrium. The social planner ignores the nominal inertia and all other
inefficiencies and chooses real allocations that maximize the representative consumer’s
utility subject to the aggregate resource constraint, the aggregate production function,
and the law of motion for habit-adjusted consumption:
max

{Xt∗ ,Ct∗ ,Nt∗ }

E0

s.t. Yt∗ = Ct∗

∞


β t u (Xt∗ , Nt∗ )

t=0

Yt∗ = At Nt∗
∗
Xt∗ = Ct∗ − θCt−1

The optimal choice implies the following relationship between the marginal rate of substitution between labor and habit-adjusted consumption and the intertemporal marginal
rate of substitution in habit-adjusted consumption

 ∗ −σ 
Xt+1
χ (Nt∗ )υ
.
−σ = At 1 − θβEt
∗
Xt∗
(Xt )
The steady state equivalent of this expression can be written as,
χ (N ∗ )υ+σ [(1 − θ) A]σ = A (1 − θβ) .

(65)

The dynamics of this model are driven by technology shocks to the system of equilibrium conditions composed of the Euler equation, the resource constraint, and the evolution
of habit-adjusted consumption. In log-linear form, these are:

1 − θβ 
∗
t∗ = θβEt X
t+1
t∗ + A
t
−υ N
X
+
σ
t + N
t∗
Yt∗ = A


∗ = 1
 ∗ − θY ∗ ,
X
Y
t
t−1
1−θ t

which combined yield the following dynamic equation
∗
∗
ζ Yt∗ = θβEt Yt+1
+ θYt−1
+



where ζ ≡ 1 + θ2 β + υδ and δ ≡

σ
(1−θβ)(1−θ) .



1+υ
δ



t
A

In the absence of deep habits, θ = 0, the


1+υ
t .
model reduces to the basic New Keynesian model where Yt∗ = σ+υ
A
44

F

Derivation of Welfare

Individual utility in period t is

N 1+υ
Xt1−σ
− t
1−σ
1+υ

where Xt = Ct − θCt−1 is the habit-adjusted aggregate consumption. Before considering
the elements of the utility function, we need to note the following general result relating
to second order approximations

where Yt = ln

 Yt 
Y

Yt − Y
1
= Yt + Yt2 + O[2]
Yt
2

and O[2] represents terms that are of order higher than 2 in the

bound on the amplitude of the relevant shocks. This will be used in various places in the
derivation of welfare. Now consider the second order approximation to the first term,
Xt1−σ
1−σ
=X
1−σ



Xt − X
X



σ 1−σ
− X
2



Xt − X
X

2

+ tip + O[2]

where tip represents ‘terms independent of policy’. Using the results above this can be
rewritten in terms of hatted variables


1
Xt1−σ
1−σ
2
t + (1 − σ)X

X
=X
+ tip + O[2].
t
1−σ
2

In pure consumption terms, the value of Xt can be approximated to second order by:
t =
X

1
1−θ



To a first order,




1 2
2
t + 1 C
t2 − θ
t−1 + 1 C
t−1
C
C
+ O[2]
− X
2
1−θ
2
2 t
t =
X

which implies

2 =
X
t

Therefore,
Xt1−σ
1−σ
=X
1−σ



1
1−θ



1 
θ 
Ct −
Ct−1 + O[1]
1−θ
1−θ


2
1
t − θC
t−1 + O[2]
C
(1 − θ)2





1
 2 + tip + O[2]
t + 1 C
2 − θ
t−1 + 1 C
2
(−σ)
X
C
C
+
t
2 t
1−θ
2 t−1
2

Summing over the future,
∞

t=0

β

1−σ
t Xt

1−σ

=X

1−σ

∞

t=0

β

t



1 − θβ
1−θ



45



1
1
2
2
t + C
 − σX

C
+ tip + O[2].
t
2 t
2

The term in labour supply can be written as


1
Nt1+υ
1+υ
2


=N
Nt + (1 + υ) Nt + tip + O[2]
1+υ
2
Now we need to relate the labor input to output which, in this case without price
dispersion, is simply,
Nt =

Yt
At

and can be approximated to first order,
t = Yt − A
t
N

which implies


2
 2 = Yt − A
t
N
t

so we can then write



Nt1+υ
1
1+υ
2




=N
Yt + (1 + υ) Yt − (1 + υ) Yt At + tip + O[2]
1+υ
2
Welfare is then given by
Γ0 = X

1−σ

E0

∞


β

t=0

−N

1+υ

E0

t



1 − θβ
1−θ





1 2
1 ˆ2

Ct + Ct − σ Xt
2
2



1
t
β t Yt + (1 + υ) Yt2 − (1 + υ) Yt A
2
t=0

∞


+tip + O[2]
From the social planner’s problem we know, X
(1 − θ)N

1+υ

−σ

υ

(1− θβ) = N such that X

1−σ

(1− θβ) =

. If we use the appropriate subsidy to render the steady-state efficient and

also use the second order approximation to the national accounting identity,
t + 1 C
 2 = Yt + 1 Yt2 − ϕ π
C
2 + O[2],
2 t
2
2 t

we can eliminate the level terms and write the sum of discounted utilities as:
!
2

∞
1 1+υ  t σ (1 − θ)  2
1
+
υ
2
t + ϕ
β
E0
Γ0 = − N
π t + tip + O[2] (66)
X + υ Yt −
A
2
1 − θβ t
υ
t=0

The welfare function can be also expressed in the usual “gap” form. To do so, we
employ the social planner’s solution in log-linear form to re-write the output term in the
46

welfare function as,


1+υ 
At
υ Yt −
υ

2

2

= υ Yt − Yt∗ + 2

Summing across time periods, we have
E0

∞

t=0


σ   ∗
∗
Yt Xt − θβEt X
t+1 + tip
1 − θβ



 
∞

2

1+υ  2
σ   ∗
t
∗
∗




β υ Yt −
At = E0
Yt Xt − θβEt Xt+1 +tip+O [2]
β υ Yt − Yt
−2
υ
1 − θβ
t

t=0

Then note that we can write

2

 2
2 = X
t X
t − X
 ∗ + 2X
∗ − X
∗
X
t
t
t
t

and, keeping only the terms relevant for policy, the welfare function becomes,
⎧
⎪
⎪
⎪
⎨

∞

σ(1−θ)
1−θβ

1 1+υ  t
Γ0 = − N
β
E0
⎪
2
⎪
t=0
⎪
⎩

Finally, we can show that

2
2


t X
t∗ + υ Yt − Yt∗
t − X
t∗ + 2 σ(1−θ) X
X
1−θβ


σ 
 ∗ − θβEt X
∗
Yt X
π 2t
−2 1−θβ
t
t+1 + ϕ

⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭

+tip+O[2]



∞



σ
σ
(1
−
θ)
∗
t X
t∗ − θβEt X
t∗ + tip
t+1
βt
= E0
βt
Yt X
X
E0
1
−
θβ
1
−
θβ
t=0
t=0
∞


which allows to write the welfare function in gap form as follows
∞

1 1+υ  t
β
E0
Γ0 = − N
2
t=0

F.1



2
2

σ (1 − θ)  
t∗ + υ Yt − Yt∗ + ϕ
π 2t
Xt − X
1 − θβ



+ tip + O[2]

Welfare Measure

We measure welfare as the unconditional expectation of lifetime utility, approximated as
W

= E

∞

t=0

=

1
1−β

β t u (Xt , Nt )




 

 

1 




u+
ΛC + ΛC−1 var Yt + 2ΛCC−1 cov Yt , Yt−1 + ΛN var Nt + Λπ var (
πt )
2

47

with u the steady-state level of the momentary utility and the Λ coefficients defined as
⎧


1−σ
σ
1
⎪
X
1
−
≡
Λ
⎪
C
⎪
1−θ
1−θ
⎪
⎪
⎪
⎪
⎪
⎨


1−σ
θ
σθ
ΛC−1 ≡ − 1−θ
X
1 + 1−θ
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ ΛCC−1 ≡ σθ 2 X 1−σ
(1−θ)

and

⎧
1+υ
⎪
⎨ ΛN ≡ − (1 + υ) N
⎪
⎩

Λπ ≡ −ϕX

1−σ

The welfare terms associated with the Ramsey policy that are involved in computing
the welfare costs of alternative policies, as in expression (28) in the text, are given by,
WXR

and

 R 1−σ
Xt
β
= E
1−σ
t=0
!
 1−σ

 


 

X
1 
1
R
+ Λπ var π
R
ΛC + ΛC−1 var YtR + 2ΛCC−1 cov YtR , Yt−1
+
=
t
1−β
1−σ
2
∞


t

WNR = −E

∞

t=0




R 1+υ
1
t Nt
β
=
1+υ
1−β

48

!
 1+υ


N
1
tR
+ ΛN var N
−
1+υ
2

G

Optimal Policy: Commitment

Upon substitution of the habit-adjusted consumption term, the central bank’s objective
function becomes



1
2
t + ϕ
− 2 (1 + υ) Yt A
π 2t
β t (δ + υ) Yt2 − 2θδ Yt Yt−1 + θ2 δ Yt−1
ΩE0
2
∞

t=0

where Ω ≡ N

1+υ

and δ ≡



1+θ
1−θ



σ
(1−θβ)(1−θ)

Yt =

and we re-write the constraints as,

1
θ 
1
1
Et Yt+1 + Et π
t+1 +
Yt−1 − R
t
1−θ
σ
1−θ
σ

π
t = βEt π
t+1 − κ2 Yt + κ2 Yt−1 − κ1 ω
t

t
μω
ω t = μωθβEt ω
 t+1 − γ 1 βEt Yt+1 + γ 2 Yt + γ 3 Yt−1 + (1 + υ) A

where

⎧
⎪
⎨ κ1 ≡
⎪
⎩

L = E0

∞


κ2 ≡

βt

t=0

⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨

⎧
σθ
⎪
γ 1 ≡ μω 1−θ
⎪
⎪
⎪
⎪
⎪
⎪
⎨


σ
γ 2 ≡ γ 1 β (1 + θ) − 1−θ
+υ
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ γ ≡ σθ (1 − μωθβ)
3
1−θ

1
ϕ

and
θ
κ1 1−θ

1
2Ω




2 + 2 (1 + υ) Y
t A
t − ϕ
− (δ + υ) Yt2 + 2θδYt Yt−1 − θ 2 δ Yt−1
π 2t




1+θ 
1 
θ 
t
−χt 1−θ
Yt − 1−θ
Yt+1 − σ1 π
t+1 − 1−θ
Yt−1 + σ1 R



t − β
π t+1 + κ2 Yt − κ2 Yt−1 + κ1 ω
t
−ψ t π

⎪
⎪
⎪
⎪



⎪
⎪
⎩ −ς t μω
t
ω t − μωθβEt ω
 t+1 + γ 1 βEt Yt+1 − γ 2 Yt − γ 3 Yt−1 − (1 + υ) A

⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭

t , Yt , π
t , and ω
 t . The first order condition with
The government chooses paths for R
respect to the nominal interest rate gives:

−σ −1 E0 β t χt = 0,

∀t ≥ 0

(67)

which implies that the IS curve is not binding and it can therefore be excluded from the
optimization problem. Once the optimal rules for the other variables have been obtained,
we use the IS curve
the path of the nominal interest rate. So, the central
" to determine
#
bank now chooses Yt , π
t , ω
 t . The Lagrangian takes the form:
L = E0

∞

t=0

βt

⎧
⎪
⎪
⎪
⎨

1
2Ω




2 + 2 (1 + υ) Y
t A
t − ϕ
− (δ + υ) Yt2 + 2θδYt Yt−1 − θ 2 δ Yt−1
π 2t



t − β
π t+1 + κ2 Yt − κ2 Yt−1 + κ1 ω
t
−ψ t π

⎫
⎪
⎪
⎪
⎬

⎪


⎪
⎪
⎪
⎪
t ⎪
⎩ −ς t μω
⎭
ω t − μωθβEt ω
 t+1 + γ 1 βEt Yt+1 − γ 2 Yt − γ 3 Yt−1 − (1 + υ) A
49

.

The first order condition for the shadow value ω
 t gives the relationship between the two

Lagrange multipliers,

κ1 ψ t = −μω (ς t − θς t−1 )

while for inflation we have the rather usual expression
π
t = −


1 
ψ t − ψ t−1 .
ϕΩ

The first order condition for output gives




t −κ2 ψ t +γ 2 ς t −γ 1 ς t−1 +βEt ΩδθYt+1 + κ2 ψ t+1 + γ 3 ς t+1 = 0
−Ωδζ Yt +Ωδθ Yt−1 +Ω (1 + υ) A


where, as defined before, ζ ≡ 1 + θ2 β + υδ .

Under full commitment, the central bank ignores past commitments in the first period
by setting all pre-existing conditions to zero, Y−1 = 0 and ψ −1 = ς −1 = 0. To find the

solution, solve the system of equations composed of the first order conditions, the three
constraints, and the technology process.

G.1

Optimal Policy: Discretion

In order to solve the time-consistent policy problem we employ the iterative algorithm of
Soderlind (1999), which follows Currie and Levine (1993) in solving the Bellman equation.

The per-period objective function can be

 written in matrix form as Zt QZt , where Zt+1 =
t+1 Yt Et Yt+1 Et ω
and
A
 t+1 Et π
t+1

⎡

0

0

⎢
0
θ2 δ
⎢
1 ⎢
Q = Ω⎢
− (1 + υ) −θδ
2 ⎢
⎢
0
0
⎣
0
0

− (1 + υ) 0 0
−θδ
(δ + υ)
0
0

⎤

⎥
0 0 ⎥
⎥
0 0 ⎥
⎥
⎥
0 0 
0 ϕ

and the structural description of the economy is given by,
Zt+1 = AZt + But + ξ t+1 ,





−1
t , ξ t+1 = εA
, A ≡ A−1
where ut = R
0 A1 , B ≡ A0 B0 ,
t+1 0 0 0 0
⎡

1 0

0

⎢
⎢ 0 1 0
⎢
A0 = ⎢
⎢ 0 0 γ 1β
⎢
1
⎣ 0 0 1−θ
0 0 0

0

0

0

0

0

0

0

σ −1

0

β

⎤

⎥
⎥
⎥
⎥,
⎥
⎥


⎡

ρ

0

⎢
0
⎢ 0
⎢
⎢
A1 = ⎢ 1 + υ
γ3
⎢
θ
− 1−θ
⎣ 0
0
−κ2
50

0

1
γ2
1+θ
1−θ
κ2

0

0

⎤

⎥
0 ⎥
⎥
−μω 0 ⎥
⎥
⎥
0
0 
κ1 1
0

and
B0 =



0 0 0 σ −1 0




.

This completes the description of the required inputs for Soderlind (1999)’s Matlab code
which computes optimal discretionary policy.

51

10

π

10

y

φ

φy

0
φ

10

π

φy

φ

φy

y

φ

0
φ

10

0
φ

10

10
0
−10
−10

0
φ

0
φ

π

0
φ

10

0
φ

Additional Figures

52

10

10

π

θ =0.9 and φR =1.1

10

10
0
−10
−10

0
φ

π

Figure 11: Determinacy properties of the model with internal habits, when monetary
t = φπ π
t−1 : determinacy (light grey dots), indepolicy follows the rule R
t + φy Yt + φR R
terminacy (blanks), and instability (dark grey stars).

H

10

θ =0.8 and φR =1.1
10
0
−10
−10

θ =0.9 and φR =0.9

10

π

0
φπ

π

π

10
0
−10
−10

10

θ =0.7 and φR =1.1
φy

y

φ

0
φ

θ =0.9 and φR =0.5
10
0
−10
−10

10

θ =0.8 and φR =0.9
10
0
−10
−10

π

θ =0.9 and φR =0

0
φπ

0
φπ
θ =0.4 and φR =1.1

10
0
−10
−10

π

y

y

φ

10
0
−10
−10

π

10
0
−10
−10

10

θ =0.8 and φR =0.5
φy

0
φ

0
φ

10

θ =0.7 and φR =0.9
10
0
−10
−10

π

θ =0.8 and φR =0
10
0
−10
−10

10

φ

0
φ

10
0
−10
−10

0
φπ

10
0
−10
−10

θ =0.4 and φR =0.9
10
0
−10
−10

θ =0.7 and φR =0.5
φy

φ

y

θ =0.7 and φR =0
10
0
−10
−10

0
φπ

θ =0 and φR =1.1

φy

10

10
0
−10
−10

y

0
φπ

10

θ =0.4 and φR =0.5
φy

φ

y

θ =0.4 and φR =0
10
0
−10
−10

0
φπ

10
0
−10
−10

φy

10

θ =0 and φR =0.9

φ

0
φπ

10
0
−10
−10

y

θ =0 and φR =0.5
φy

φ

y

θ =0 and φR =0
10
0
−10
−10

10

1.2

35

1

φ

π

30

φ

R

0.8

0.2
15

y

φ

π

0.4
20

R

0.6
φ and φ

25

0
φ

y

−0.2
10
−0.4
5
0.1

0.2

0.3

0.4

0.5
θ

0.6

0.7

0.8

−0.6
0.9

Figure 12: Optimal policy rule parameters for varying degrees of superficial habits, the
unconstrained case.

53

50

4

45

3.5

φπ

φR

40

3

35

1.5

y

φ

π

2
25

φ and φ

30

R

2.5

20
1

φy

15

0.5

10

0

5
0
0.1

0.2

0.3

0.4

0.5
θ

0.6

0.7

0.8

−0.5
0.9

Figure 13: Optimal policy rule parameters for varying degrees of deep habits, the unconstrained case.

54

Inflation
0.1

0

0.05

%

0.05

−0.05

%

Output gap

0

2
4
Nominal interest rate

0

6

0

0

−0.1

−0.1

−0.2

0

2

4

−0.2

6

0

0

2
4
Real interest rate

2

%

Output
0.5

0.2

0

0

2

4

6

4

6

Markup

0.4

0

6

4

−0.5

6

years

0

2
years

Figure 14: Impulse responses to a 1% positive technology shock in the model with deep
habits with θ = 0.6, under the optimal Taylor rule (dash lines) and optimal commitment
policy (solid lines).

55