A Comparison of 2 popular models of monetary policy
Petros Varthalitis
Athens University of Economics & Business
June 2011
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
1 / 45
Aim of this work
Obviously, CB’s need models with nominal rigidities which allow a real role
for monetary policy.
1. Calvo model (Staggered Pricing, Calvo 1983 JME)
2. Rotemberg model (Rotemberg 1982 JPE)
The aim of this study is to compare them. This is interesting both in:
Policy Level: Lombardo & Vestin ECB wp & EL (2008).
Theory Level: Ascari & Rossi (2010) wp, Dellas & Collard (2007)
J.Macro
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
2 / 45
Key Features of the model (in words):
1. Households consume, work, save in money, private bonds & capital.
2. Government …nances government expenditures with seignorage
revenues and lump-sum taxes.
3. Firms operate under Monopolistic Competition, i will examine two
models of price setting:
a. Calvo
b. Rotemberg
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
3 / 45
Household’s Problem (both models)
indexed by j
∞
∑ βt U (ct (j ), mt (j ), nt (j ), gt (j ))
max
fct (j ),ct (i ,j ),xt (i ,j ),m t (j ),n t (j ),kt (j ),b t (j )gt∞=0 t =0
(1)
subject to:
= Rt
ct (j ) + xt (j ) + bt (j ) + mt (j )
Pt 1
Pt 1
k
b
(
j
1 Pt
t 1 ) + P t mt 1 (j ) + wt nt (j ) + rt kt 1 (j )
kt (j ) = (1
Varthalitis (AUEB)
δ) kt
1
τ lt (j )
(j ) + xt (j )
Calvo vs Rotemberg
(2)
(3)
June 2011
4 / 45
Household’s Problem (both models)
There are i di¤erentiated goods.Using the DS aggregator we have:
Varthalitis (AUEB)
ct ( j )
Z 1
xt (j )
Z 1
0
0
ct (i, j )
ε 1
ε
xt (i, j )
ε 1
ε
ε
ε 1
(4)
di
ε
ε 1
(5)
di
Pt ct (j ) =
Z 1
Pt (i ) ct (i, j ) di
(6)
Pt xt (j ) =
Z 1
Pt (i ) xt (i, j ) di
(7)
0
0
Calvo vs Rotemberg
June 2011
5 / 45
Households- FOC (both models)
[Euler for Bonds] :
1
ct +1 (j ) σ Pt
= βEt
Rt
ct (j ) σ Pt +1
(8)
[Euler for Capital] :
ct ( j )
[Money Demand] :
σ
h
= βEt (1
i
δ) + rtk+1 ct +1 (j )
σ
(9)
Rt 1
[mt (j )] ν
=
σ
ct ( j )
Rt
(10)
nt (j )φ
= wt
ct ( j ) σ
(11)
[Labour Supply] :
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
6 / 45
Households- FOC (both models)
[Demand for i ] :
Pt (i )
Pt
ct (i, j ) + xt (i, j ) =
where Pt =
R1
0
Varthalitis (AUEB)
Pt (i )1
ε
di
1
1 ε
ε
fct (j ) + xt (j )g
(12)
.
Calvo vs Rotemberg
June 2011
7 / 45
Government (both models)
Government Budget Constraint:
Pt τ lt + Mt = Mt
where
gt
Varthalitis (AUEB)
Z 1
0
gt (i )
1
ε 1
ε
Calvo vs Rotemberg
+ Pt gt
(13)
ε
ε 1
di
(14)
June 2011
8 / 45
Firms
Step A: Cost Minimization (both models)
Firms i 2 [0, 1] .
Perfect Competition in factor markets.
Ψt (Yt (i )) =
min
fn t (i ), k t
1 (i )g
subject to:
Yt (i ) = At kt
Varthalitis (AUEB)
n
Wt nt (i ) + Rtk kt
1 (i )
a
nt (i )1
Calvo vs Rotemberg
1 (i )
o
(15)
a
(16)
June 2011
9 / 45
Firms- Calvo Model
1
2
Monopolistic Competition. Yt (i ) , i 2 [0, 1] .
In Calvo model each period t there are two fraction of …rms:
θC
1 θC
Cannot Reoptimize
Reoptimize
where θ C is an exogenous probability.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
10 / 45
Firms- Calvo Model
A …rm which cannot reoptimize (belongs to θ C ) just set its previous
period price:
Pt (i ) = Pt 1 (i )
(17)
θ C is Calvo nominal rigidity parameter.
In Calvo model there is heterogeneity across …rms. Symmetry fails.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
11 / 45
Firms- Calvo Model
Step B: Pro…t Maximization
∞
∑
(i )
max
Pt
θC
k =0
k
Et fQt,t +k (Pt (i )Yt +k (i )
Ψt +k (Yt +k (i )))g
(18)
subject to:
Yt +k (i ) =
Pt (i )
Pt +k
ε
Ytd+k
(19)
and given Ytd
Ct + Xt + Gt is aggregate demand, Qt,t +k is the
stochastic discount factor and Ψt (..) is the minimum cost function.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
12 / 45
Firms
Calvo Price Setting Rule
∞
∑
θC
k
Et
k =0
(
Qt,t +k
Pt (i )
Pt +k
ε
Ytd+k
Pt (i )
ε
ε
1
Ψt0 +k
)
=0
(20)
Firm i sets its price Pt (i ) at period t as the weighted sum of the
expected nominal marginal costs of the next k periods.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
13 / 45
Firms- Rotemberg Model
1
2
Monopolistic Competition. Yt (i ) , i 2 [0, 1] .
Each …rm i faces a convex price adjustment cost. All …rms solve an
identical problem each period:
PAC =
ϑR
2
Pt (i )
Pt 1 (i )
2
1
Yt (i )
(21)
where ϑR measures nominal price rigidity. As ϑR increases so does nominal
price rigidity.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
14 / 45
Firms- Rotemberg Model
In Rotemberg model, all …rms i solve an identical problem. So, there
is symmetry.
ϑR is Rotemberg nominal rigidity parameter.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
15 / 45
Firms- Rotemberg Model
Step B: Pro…t Maximization
∞
max
8
<
∞
∑ Qt,t +1 Ωe t = ∑ Qt,t +1 :
fP t (i )g t =0
t =0
P t (i )
P t Yt (i )
P t (i )
ϑR
2
P t 1 (i )
subject to:
Yt (i ) =
Varthalitis (AUEB)
Pt (i )
Pt
Calvo vs Rotemberg
9
Ψrt (Yt (i )) =
1
2
Yt (i ) ;
(22)
ε
Yt
(23)
June 2011
16 / 45
Firms - Rotemberg
Rotemberg price setting rule
Qt,t +1 (1
= Qt ϑR
ε) Yt + Qt εΨrt 0 (.) Yt + Qt,t +2 ϑR
Pt
Pt 1
Varthalitis (AUEB)
1
Pt +1
Pt
1
Pt +1
Pt
Pt
Pt 1
(24)
Calvo vs Rotemberg
June 2011
17 / 45
Aggregation (in words)
In Calvo model there is an aggregation issue which arises from the
presence of two fraction of …rms in each period t, θ C which cannot
reoptimize and 1 θ C which reoptimize.
In Rotemberg model, …rms are symmetric and aggregation is trivial.
(Next slides appendix)
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
18 / 45
Calvo Model: Aggregation
Household aggregation is trivial:
xt
where xt (j ) =
and xt =
ct
Z 1
0
xt (j ) dj
(25)
e t (j ) bt (j ) mt (j )
ct (j ) xt (j ) kt (j ) nt (j ) Ω
0
e t bt mt .
xt kt nt Ω
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
0
19 / 45
Calvo Model: Aggregation
Aggregate Supply:
Yts
Z 1
0
Yt (i )
ε 1
ε
ε
ε 1
(26)
di
Each …rm i faces an identical technology:
Yt (i ) = At kt
Varthalitis (AUEB)
1
(i )a nt (i )1
Calvo vs Rotemberg
a
(27)
June 2011
20 / 45
Calvo Model: Aggregation
Aggregation:
Yts
Z 1
0
a
At kt (i ) nt (i )
1 a
ε 1
ε
ε
ε 1
di
=? = At kta 1 nt1
a
(28)
R1
0
We denote two auxiliary indices Yt
At kt (i )a nt (i )1 a di and
0
1
R1
0
ε
Pt
P i ε di
. We can proove that (an analytical appendix will
0 t ( )
be available):
1
Yts = h 0 i ε At kta nt1 a
(29)
Pt
Pt
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
21 / 45
Calvo Model: Price Dispersion
Yun (1996) denotes:
∆t
"
0
Pt
Pt
#
ε
(30)
This is a measure of price dispersion and measures the loss of output in
the Calvo economy due to price dispersion, ∆t
1 where equality holds
Pt
only when Pt 1 = 1.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
22 / 45
Calvo Model: Evolution of Aggregate Price Level
Pt1
ε
= θPt1
ε
1
+
Z
Sθ
(Pt (i ))1
ε
di = θPt1
ε
1
+ (1
θ ) (Pt )1
ε
(31)
All …rms i which reoptimize at period t solves an identical problem so
Pt (i ) = Pt .
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
23 / 45
Aggregation Rotemberg
No aggregation problems in Rotemberg model. Households (j) are
symmetric, so:
Z
1
xt =
0
xt (j ) dj
(32)
Firms i are symmetric, i.e. aggregate supply:
Z 1
Yts
=
Yt (i )
0
At kta 1 nt1 a
Varthalitis (AUEB)
ε 1
ε
ε
ε 1
di
=
Z 1
0
a
At kt (i ) nt (i )
1 a
ε 1
ε
ε
ε 1
di
(33)
Calvo vs Rotemberg
June 2011
24 / 45
Transformations
In Calvo model we de…ne 3 new endogenous variables which subsitute the
0
0
:
price levels Pt Pt Pt
Θt
∆t
Πt
Pt
Pt
" 0#
Pt
Pt
(34)
ε
(35)
Pt
Pt 1
(36)
In Rotemberg model we only subsitute the aggregate price level with
in‡ation:
Πt
Varthalitis (AUEB)
Pt
Pt 1
Calvo vs Rotemberg
(37)
June 2011
25 / 45
Decentralized Equilibrium Calvo Model (given policy)
13 endogenous variables:
Yt ct kt nt xt wt rtk
for 13 equilibrium equations.
Given policy
Varthalitis (AUEB)
Rt
stg
0
mt
stl
mct
Πt
0
and
∆t
Θt
0
, and an exogenous shock At .
Calvo vs Rotemberg
June 2011
26 / 45
Decentralized Equilibrium Calvo Model
ct
σ
= βEt ct +1
σ
h
rtk+1 + (1
ct σ
= βEt ct +1
Rt
mt
ct
ν
σ
=
σ
δ)
i
(38)
1
Π t +1
(39)
Rt 1
Rt
(40)
nt φ
= wt
ct σ
Varthalitis (AUEB)
Calvo vs Rotemberg
(41)
June 2011
27 / 45
Decentralized Equilibrium Calvo Model
kt = (1
wt = mct (1
δ) kt
a)At kt
rtk = mct aAt kta
τ lt + mt = mt
Varthalitis (AUEB)
1
1
+ xt
1
a
nt
(42)
a
(43)
1 1 a
1 nt
(44)
1
+ gt
Πt
(45)
Calvo vs Rotemberg
June 2011
28 / 45
Yt = ct + xt + gt
∞
∑
k =0
θ
C
k
Et
8
>
>
>
>
<
>
>
>
>
:
Qt,t +k
2
6
6
6
6
4
3
Θt
k
∏ Π t +i
i =1
Yt =
[ Πt ]1
ε
7
7
7
7
5
Yt +k
0
B
@
1
At kta 1 nt1
∆t
= θ C + (1
∆t = θ∆t
Varthalitis (AUEB)
ε
ε
1 Πt
(46)
Θt Πt
k
ε
ε 1 MCt +k
Calvo vs Rotemberg
i =0
(47)
a
(48)
θ C ) [ Θt Πt ]1
+ (1
∏ Π t +i
9
1>
>
>
>
=
C
A =0
>
>
>
>
;
θ ) Θt
ε
(49)
ε
(50)
June 2011
29 / 45
Decentralized Equilibrium Rotemberg Model (given policy)
11 endogenous variables:
Yt ct kt nt xt wt rtk
for 11 equilibrium equations.
Given policy
Varthalitis (AUEB)
Rt
stg
0
mt
stl
mct
Πt
0
, and an exogenous shock At .
Calvo vs Rotemberg
June 2011
30 / 45
Decentralized Equilibrium Rotemberg Model
ct
σ
= βEt ct +1
σ
h
rtk+1 + (1
ct σ
= βEt ct +1
Rt
mt
ct
ν
σ
=
σ
δ)
i
(51)
1
Π t +1
(52)
Rt 1
Rt
(53)
nt φ
= wt
ct σ
Varthalitis (AUEB)
Calvo vs Rotemberg
(54)
June 2011
31 / 45
Decentralized Equilibrium Rotemberg Model
kt = (1
wt = mct (1
δ) kt
a)At kt
rtk = mct aAt kta
τ lt + mt = mt
Varthalitis (AUEB)
1
1
+ xt
1
a
nt
(55)
a
(56)
1 1 a
1 nt
(57)
1
+ gt
Πt
(58)
Calvo vs Rotemberg
June 2011
32 / 45
Decentralized Equilibrium Rotemberg Model
Yt = ct + xt + gt +
ϑR
( Πt
2
Yt = At kta 1 nt1
Qt,t +1 (1
R
(59)
a
ε) Yt + Qt,t +1 εmct Yt + Qt,t +2 ϑR (Πt +1
= Qt,t +1 ϑ (Πt
Varthalitis (AUEB)
1)2 Yt
(60)
1) Πt +1 Yt
1) Πt Yt
Calvo vs Rotemberg
(61)
June 2011
33 / 45
Comparison of Calvo versus Rotemberg (Nominal Rigidity)
The di¤erent price setting mechanism causes:
Price dispersion in Calvo model.
A (price adjustment) cost in Rotemberg model.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
34 / 45
Comparison of Calvo versus Rotemberg DEs
1. Steady-State
2. Transition (Dynamics up to a …rst-order approximation).
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
35 / 45
Steady-state with zero long-run in‡ation
Calvo
If Π = 1, we can proove that:
∆=Θ=1
(62)
PAC = 0
(63)
So price dispersion vanishes.
Rotemberg
So the price adjustment cost is zero.
The two steady-state solutions are identical.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
36 / 45
Dynamics around zero in‡ation steady-state
Production function in Calvo:
bt = A
b t + akbt
Y
1
+ (1
a) b
nt
bt
∆
(64)
b t = θ∆
b t 1 is a deterministic univariate autoregressive process
However ∆
which does not a¤ect the dynamics of the other endogenous variables of
the model.
In Rotemberg model the price adjustment cost is zero both o¤ and in
steady state so the …rst-order approximation of the resource constraint is
identical in two models.
The equilibrium relations of the two models (except from the price setting
rules)
are equivalent up to a …rst-order approximation.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
37 / 45
In‡ation Dynamics around zero in‡ation steady-state
The linear New Keynesian Phillips Curves:
Calvo:
Rotemberg:
b t = βEt Π
b t +1 + λC mc
Π
ct
(65)
b t = βEt Π
b t +1 + λR mc
Π
ct
(66)
where λC and λR depend on structural parameters of each model.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
38 / 45
Condition for equivalent Dynamics around zero in‡ation
steady-state
The two models deliver equivalent dynamics up to a …rst-order when:
λC = λR
1
θC
1
θ
Calvo
θ C (Nominal Rigidity)
β (Time preference)
(67)
βθ C
=
C
1
ε
ϑ
(68)
R
Rotemberg
ϑR (Nominal Rigidity)
ε (Price elasticity of demand)
The slope of Linear NKPC depends on the nominal rigidity parameters and
on DIFFERENT structural parameters of the models
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
39 / 45
Slope of NKPC for equivalent dynamics
θ C implies a ϑR such that λC = λR , given β and ε :
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
40 / 45
A Sensitivity analysis of a non-zero in‡ation steady-state
Output with respect to In‡ation
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
41 / 45
A Sensitivity analysis of a non-zero in‡ation steady-state
Consumption with respect to In‡ation
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
42 / 45
A Sensitivity analysis of a non-zero in‡ation steady-state
Output with respect to Nominal Rigidity Parameter
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
43 / 45
Sum up:
1
Rotemberg is a simpler model (at least in algebra).
2
Zero long-run in‡ation generates: i. identical steady-state solution ii.
Equivalent dynamics given that the nominal rigidity parameters satisfy
a speci…c condition.
3
Non- zero long-run in‡ation generates important di¤erences between
the two models.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
44 / 45
How we will proceed
1. Which model does better vis-à-vis the data (Calibration).
2. Macroeconomic & Welfare Implications of di¤erent policy rules.
Varthalitis (AUEB)
Calvo vs Rotemberg
June 2011
45 / 45