Microfoundations of DSGE Models: III Lecture
B. Annicchiarico
Barbara Annicchiarico
BBLM del Dipartimento del Tesoro
21 Giugno 2010
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
1 / 65
Contents
A New Keynesian model with capital accumulation
A New Keynesian model with capital accumulation, habit
persistence and adjustment costs on labour and investments
A New Keynesian model with capital accumulation and
rule-of-thumb households
A complete model
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
2 / 65
Model #1: A NK Model with Capital Accumulation
Main Features
Agents
a continuum of households that consume, own capital and supply
differentiated labour services;
a continuum of trade unions each of which representing workers of
a certain type;
a continuum of intermediate goods producers that employ labour
inputs, rent capital from households and produce differentiated
intermediate goods (monopolistic competition);
a continuum of final goods producers that use intermediate goods
to produce a homogenous final good consumed by households
(perfect competition);
the government that set public spending;
the central bank that implements monetary policy.
Imperfect competition
Prices rigidities
Business cycles driven by nominal and real shocks
Rational expectations and no asymmetric information
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
3 / 65
Model #1: A NK Model with Capital Accumulation
Final-Good Firms
Each firm takes the other firms’ prices as given.
The representative final-good firm uses Yt (j ) units of each
intermediate good j 2 [0, 1] purchased at a nominal price Pt (j ) to
produce Yt units of the final good with a constant returns to scale
technology:
Yt =
Z 1
0
Yt (j )
θ 1
θ
θ
θ 1
dj
θ = the elasticity of substitution across intermediate goods, θ > 1. As
θ ! ∞ higher and higher degree of substitution ! less market power
of intermediate-goods producers.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
4 / 65
Model #1: A NK Model with Capital Accumulation
Final-Good Firms
The problem of the representative firm is to max their profits wrt Yt (j ) with
j 2 [0, 1] (static problem) given the available technology.
Z 1
Pt Yt
0
Pt (j )Yt (j ) dj
Profit maximization yields the following set of demands for intermediate
goods:
Yt (j ) =
θ
Pt (j )
Pt
Yt
Perfect competition and free entry drives the final good-producing firms’
profits to zero, so that from the zero-profit condition we obtain:
Pt =
Z 1
0
Pt (j )
1 θ
1
1 θ
dj
.
which defines the aggregate price index of the economy and is such that
Pt Yt =
B. Annicchiarico
R1
0
Pt (j ) Yt (j ) dj .
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
5 / 65
Model #1: A NK Model with Capital Accumulation
Intermediate-Goods Firms
Each intermediate good j is produced by a monopolist which has a
production function of the form:
Yt (j ) = At Lt (j )α Kt (j )1
α
where 0 < α < 1
At = Total factor productivity, At = A exp(εA,t ) and εA,t = ρA εA,t + ξ A,t
with ξ A,t
iid.N (0, σ2A )
Lt (j ) = CES aggregate of labor inputs supplied by unionized workers
defined below (see below)
Kt (j ) = physical capital
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
6 / 65
Model #1: A NK Model with Capital Accumulation
Price Rigidities à la Rotemberg
According to Rotemberg (1983) each monopolistic firm faces a
quadratic cost of adjusting nominal prices, measured in terms of the
final-good:
2
γp
Pt (j )
1 Yt
ADJ_Pt (j ) =
2 Pt 1 (j )
where γp =degree of nominal price rigidities.
Firm j will not always choose to charge the optimal price (i.e.
Pt (j ) = θ θ 1 MCt (j )) since it is assumed to face convex costs to
changing its price.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
7 / 65
Model #1: A NK Model with Capital Accumulation
Price Rigidities à la Rotemberg
Why are price changes costly?
There is the cost of physically changing posted prices (probably a
fixed cost per price change).
A firm that changes its prices may face a cost which results from
the negative reaction of its costumers (reputation loss). From this
point of view, probably customers react more strongly to large
price changes than to gradual variations.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
8 / 65
Model #1: A NK Model with Capital Accumulation
Price Rigidities à la Rotemberg
What is the main difference between the Calvo price setting and
the Rotemberg adjustment costs hypothesis?
In the Rotemberg model there is no price dispersion. In each period
t firm j can change its price price Pt (j ) s.t. the payment of the
adjustment cost. All firms face the same problem and will set the
same price and produce the same quantity of each differentiated
good (symmetry).
In the Calvo model there’s price dispersion. Firms will not set the
same price (asymmetry).
Different types of inefficiency in the two models
B. Annicchiarico
In the Rotemberg model the cost of nominal rigidities consists in a
wedge between aggregate demand (Ct + It + Gt ) and aggregate
output (Yt ), since a fraction of the output goes in the price
adjustment cost.
In the Calvo model, nominal rigidities through price dispersion,
makes aggregate output less efficient.
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
9 / 65
Model #1: A NK Model with Capital Accumulation
Price Rigidities à la Rotemberg
Despite the economic difference between these two models of price
rigidities, up to a first order approximation and around a zero inflation
steady state, they imply the same reduced form of the New Keynesian
Phillips curve.
Otherwise, the Rotemberg model seems to be more robust to
non-linearities (implying more robust results).
The implications of of having trend inflation in the two pricing models.
On these issues: see Ascari and Merkl (2009); Ascari and Ropele (2007);
Ascari and Rossi (2009, 2010).
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
10 / 65
Model #1: A NK Model with Capital Accumulation
Intermediate-Goods Firms
Given the wage index Wt and the rental rate of capital rtk , the problem
for firm j is to choose fLt (j ) , Kt (j ), Pt (j )gt∞ = 0 in order to maximize
the sum of expected discounted real profits
)
(
R
∞
P t (j )
Wt
k K (j )+
λ
Y
j
L
j
r
(
)
(
)
t
t t
Pt
Pt t
,
E0 ∑ βt Rt
ADJ_P
(
j
)
λ
t
t =0
0
given Yt (j ) =
ADJ_Pt (j ) =
B. Annicchiarico
γp
2
P t (j )
Pt
θ
P t (j )
P t 1 (j )
Yt and where
1
2
Yt .
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
11 / 65
Model #1: A NK Model with Capital Accumulation
Intermediate-Goods Firms
To solve firm’s j problem,
8 2 consider the Lagrangian function
3
P t (j )
Wt
k K (j )
>
Y
j
L
j
r
(
)
(
)
>
t
t
t
t
Pt
< 4 Pt
5+
R
2
γp
P t (j )
L0 = E0 ∑t∞=0 βt λλRt
1
Y
t
2
P t 1 (j )
0 >
>
:
MCt (j ) Yt (j ) At Lt (j )α Kt (j )1 α
where MCt (j ) =real marginal cost.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
9
>
>
=
>
>
;
21 Giugno 2010
12 / 65
Model #1: A NK Model with Capital Accumulation
Intermediate-Goods Firms
Using Yt (j ) =
P t (j )
Pt
θ
Yt we have:
FOC wrt Pt (j )
1
(1
Pt
θ ) + MCt (j ) θ
1
Yt (j ) =
Pt (j )
∂ADJ_Pt (j )
+
∂Pt (j )
+ βEt
Remark: for γp = 0 under symmetry: MCt =
B. Annicchiarico
θ 1
θ ,
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
λRt+1 ∂ADJ_Pt +1 (j )
∂Pt (j )
λRt
that is Pt =
θ
N
θ 1 MCt .
21 Giugno 2010
13 / 65
Model #1: A NK Model with Capital Accumulation
Intermediate-Goods Firms
FOC wrt Kt (j )
rtk = (1
α) MCt (j ) At Lt (j )α Kt (j )
α
FOC wrt Lt (j )
B. Annicchiarico
Wt
= αMCt (j )At Lt (j )α
Pt
1
Kt ( j ) 1
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
α
21 Giugno 2010
14 / 65
Model #1: A NK Model with Capital Accumulation
Households and Preferences
Consider a continuum of households index by i 2 [0, 1]. Household i
is characterized by the following lifetime utility function:
∞
E0
∑ βt
log(Ci ,t ) +
t =0
ω
1
v
(1
Li ,t )1
v
Ci ,t consumption; Li ,t specific labour inputs; β is the time discount
factor; v < 1
The period-by-period budget constraint is
Pt Ci ,t + Bi ,t + Pt Ii ,t
= Wi ,t Li ,t + (1 + rt
+Pt rtK Ki ,t + Di ,t
1 )Bi ,t 1
+
Pt TAXi ,t
where Ii ,t = Ki ,t +1 (1 δ) Ki ,t ; Di ,t = dividends; Wi ,t =nominal
wage; TAXt =lump-sum taxes; r = nominal interest rate;
Bi ,t 1 =nominal bonds issued by the government.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
15 / 65
Model #1: A NK Model with Capital Accumulation
Households and Preferences
The representative household will choose fCi ,t , Bi ,t , Ki ,t +1 gt∞=0 so as to
max the lifetime utility function given the sequence of budget
constraint. To solve household’s j problem, consider the Lagrangian
function
9
1 v
ω
1
L
+
log
C
+
(
)
(
)
i
,t
i
,t
=
1
v
∞
# >
"
W i ,t
B i ,t 1
t
i
K K + D i ,t +
L
+
(
1
+
r
)
+
r
L 0 = E0 ∑ β
t 1
i ,t
t
P t i ,t
Pt
Pt
>
>
B i ,t
;
: λi ,t
t =0
K
+
1
δ
TAXi ,t Ci ,t
(
) Ki ,t
i ,t +1
Pt
8
>
<
FOC wrt Ci ,t :
1
C i ,t
λi ,t
Pt
= λi ,t
λ
FOC wrt Bi ,t :
= βEt Pi ,tt ++11 (1 + rt )
FOC wrt Ki ,t +1 : λi ,t = βEt λi ,t +1 rtK+1 + (1
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
δ)
21 Giugno 2010
16 / 65
Model #1: A NK Model with Capital Accumulation
Households and Preferences
Combining the previous conditions we derive the Euler equation
governing the time path of consumption
1
1 + rt
1
= βEt
Ci ,t
Ci ,t +1 1 + π t +1
and the asset equation according to which at the optimum a household
is indifferent between the two assets (capital and risk-free public debt
bonds) since the expected benefit in terms of utility is the same:
B. Annicchiarico
Et λi ,t +1
h
1 + rt
= Et λi ,t +1 rtK+1 + (1
1 + π t +1
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
δ)
i
21 Giugno 2010
17 / 65
Model #1: A NK Model with Capital Accumulation
Wage Setting
There is a continuum of unions each of which represents workers of a
certain type. Effective labour input hired by the intermediate-good
firm j is a CES function of the quantities of the different labour types
employed:
! σ σL 1
L
σL 1
R1
Lt (j ) =
Li ,t (i ) σL di
0
where σL > 1 elasticity of substitution across different types of labour
inputs. At the optimum (and under symmetry among firms) the
demand for each variety of labour input i is
Li ,t =
where Wt =
B. Annicchiarico
hR
1
0
Wi1,t
σL
di
i 1 1σ
L
Wi ,t
Wt
σL
Lt
such that Wt Lt =
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
R1
Wi ,t Li ,t di.
0
21 Giugno 2010
18 / 65
Model #1: A NK Model with Capital Accumulation
Wage Setting
The union representing worker of type i will set Wi ,t in order to max
the expected utility of household i given Li ,t =
relevant part of the Lagrangian is
E0 β t
ω
1
v
(1
Li ,t )1
v
+ λi ,t
W i ,t
Wt
σL
Lt . The
Wi ,t
Li ,t
Pt
At the optimum:
B. Annicchiarico
Wi ,t
=
Pt
σL
σ
1
| L{z }
ω (1
Li ,t )
v
λi ,t
wage markup
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
19 / 65
Model #1: A NK Model with Capital Accumulation
The Government
The government budget constraint is
Bt = (1 + rt )Bt
1
+ Pt Gt
Pt TAXt
where Gt = G exp(εg ,t ) and εg ,t = ρg εg ,t + ξ g ,t with ξ g ,t
while
Pt TAXt = Pt TAX + τPt Bt
iid.N (0, σ2g ),
where τ is set in order to rule out any explosive path of the public
debt ("passive" rule as meant by Leeper 1991).
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
20 / 65
Model #1: A NK Model with Capital Accumulation
The Central Bank
The monetary authority sets the short-term nominal interest rate in
accordance with an interest rate rule
Rt
=
R
Rt 1
R
ιr
Πt
Π
ιπ
Yt
Y
ιy 1 ιr
utR .
where Rt = 1 + rt , Πt = 1 + π t , utR = exp (εu,t ) and εu,t = ρu εu,t + ξ u,t
with ξ u,t
iid.N (0, σ2u ).
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
21 / 65
Model #1: A NK Model with Capital Accumulation
Equilibrium
Combining the above conditions, imposing symmetry between firms,
households and unions the equilibrium of the economy is described by
the following equations.
The Euler equation
1 + rt
λt = βEt λt +1
1 + π t +1
The capital asset equation
h
i
λt = βEt λt +1 rtK+1 + (1 δ)
The Lagrange multiplier
λt =
1
Ct
The wage equation
B. Annicchiarico
Wt
σL ω (1 Lt )
=
Pt
σL 1
λt
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
v
21 Giugno 2010
22 / 65
Model #1: A NK Model with Capital Accumulation
Equilibrium
The aggregate production function
Yt = At Ltα Kt1
α
The demand of capital
rtk = (1
α) MCt At Ltα Kt
α
The demand of labour
Wt
= αMCt At Ltα
Pt
1
Kt1
α
The inflation equation
1
γp ( Πt
B. Annicchiarico
1) Πt + βγp Et
λRt+1
λRt
( Π t +1
1)
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
Yt +1
Π t +1 = ( 1
Yt
MCt ) θ
21 Giugno 2010
23 / 65
Model #1: A NK Model with Capital Accumulation
Equilibrium
The budget constraint of the government
Bt = (1 + rt )Bt
1
+ Pt Gt
Pt TAXt
The tax rule
Pt TAXt = Pt TAX + τPt Bt
The interest rate rule
B. Annicchiarico
Rt
=
R
Rt 1
R
ιr
Πt
Π
ιπ
Yt
Y
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
ιy 1 ιr
utR .
21 Giugno 2010
24 / 65
Model #1: A NK Model with Capital Accumulation
Equilibrium
The capital accumulation equation
Kt + 1 = ( 1
δ)Kt + It
The resource constraint of the economy
Yt = Ct + It + Gt +
γp
( Πt
2
1)2 Yt
The exogenous processes governing At , Gt and utR
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
25 / 65
Model #1: A NK Model with Capital Accumulation
A remark on the inflation equation
Given the inflation equation
1
γp ( Πt
1) Πt + βγp Et
λRt+1
λRt
( Π t +1
in a zero-inflation steady state MC =
1)
Yt +1
Π t +1 = ( 1
Yt
θ 1
θ .
With trend inflation (and no indexation): MC =
The markup will be:
markup =
1
θ
θ
+ γp
(1
β) (Π
θ
θ 1
θ
+ γ p (1
1) Π
MCt ) θ
β)(Π 1 )Π
.
θ
1
The markup is decreasing in the level of trend inflation. As a result:
output (and so employment) is higher the higher Π. However, a
higher fraction of output is eaten up by the adjustment costs.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
26 / 65
Model #1: A NK Model with Capital Accumulation
Calibration
B. Annicchiarico
α = 2/3
β = 0.99
δ = 0.1/4
v = 0.8
γp = 58.25
θ=6
σL = 5
ρA = 0.9
ρG = 0.9
ρR = 0.9
labour share
discount factor
depreciation rate
preference parameter
degree of price rigidities 1 θ = 0.25
elasticity of subst. between goods
elasticity of subst. between labour inputs
persistence of tech shock
persistence of the public spending shock
persistence of monetary policy shock
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
27 / 65
Model #1: A NK Model with Capital Accumulation
Calibration
B. Annicchiarico
Π=1
L = 0.3
Y =1
ιπ = 1.5
ιR = 0
ιy = 0.5/4
inflation
employment
output
monetary policy parameter
monetary policy parameter
monetary policy parameter
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
28 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Technology Shock
B. Annicchiarico
Consumption, c
Output, Y
0.6
Labour, L
1.4
Investments, I
0.6
6
1.2
5
0.4
1
0.8
0.3
0.6
4
3
%
0.4
%
0.5
0.2
2
0.4
0.2
0.1
1
0
0.2
0
20
40
0
0
0
20
quarters
40
-0.2
0
20
40
-1
Inflation
Real Interest Rate
0.05
0.7
0.04
0.2
0.6
0.03
0.15
-0.1
0.5
0.02
0.1
-0.15
0.4
0.01
0.05
0.3
-0.2
20
quarters
40
0.1
0.25
0
0.2
0
0
-0.01
0
20
quarters
40
Public Debt, B/P
0.8
-0.05
20
quarters
Real Wage, W/P
0
-0.25
0
quarters
40
-0.02
-0.05
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
-0.1
0
20
40
quarters
21 Giugno 2010
29 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Technology Shock: Propagation
Higher Productivity ! lower marginal costs ! lower inflation
As in the basic RBC model a transitory productivity shock, which
temporarily raises the real wage rate, increases employment
(maybe we have a too low degree of price rigidity)
Productivity " !MPK " !rental rate of capital "
B. Annicchiarico
Substitution effect increases savings (prevails)
Consumption increases gradually (consumption smoothing)
Investments increases on impact (the volatile component)
As a result y increases more than proportionally.
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
30 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Technology Shock with a higher degree of price rigidities1
C ons umption, c
Output, Y
0.7
1.4
0.6
1.2
Labour, L
Inv es tments , I
0.6
6
5
0.4
0.8
0.2
4
0.3
0.6
0
0.2
0.4
0.1
0.2
3
%
1
0.4
%
0.5
0
0
20
40
0
2
1
-0.2
0
20
quarters
40
-0.4
0
0
20
40
-1
0
quarters
Inflation
0
-0.05
20
40
quarters
R eal W age, W /P
R eal Interes t R ate
0.7
0.05
0.6
0.04
0.5
0.03
-0.1
0.4
0.02
-0.15
0.3
0.01
0.2
0
Public D ebt, B/P
0.2
0.1
0
-0.2
-0.25
0
20
quarters
1 Reponses
B. Annicchiarico
40
0.1
-0.01
0
-0.02
0
20
quarters
40
-0.1
0
20
40
quarters
-0.2
0
20
40
quarters
under the baseline calibration are plotted in red.
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
31 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Government Spending Shock
B. Annicchiarico
Consumption, c
Output, Y
-0.01
Labour, L
0.08
-0.02
Investments, I
0.12
0.1
0.1
0.06
-0.03
0
0.08
0.02
%
-0.05
-0.1
%
-0.04
0.04
0.06
-0.2
0.04
-0.06
0
-0.07
-0.08
0
20
20
quarters
40
0
0
20
40
-0.4
0
quarters
-3
6
-0.3
0.02
-0.02
40
0
x 10 Inflation
Real Wage, W/P
-0.005
3
-0.01
5
20
40
quarters
-3
Real
x 10 Interest Rate
Public Debt, B/P
1
2.5
0.8
-0.015
4
-0.02
2
0.6
3
-0.025
1.5
0.4
1
0.2
-0.03
2
1
-0.035
0
20
quarters
40
-0.04
0
20
quarters
40
0.5
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
32 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Government Spending Shock: Propagation
Government spending " !taxes increase " !net wage income#
income effect (agents need to work more) !employment increases
then gradually returns to normal
consumption falls, but the rise in government supply is temporary,
hence agents respond by decreasing their capital holdings
(consumption smoothing)
firms will revise their prices upward (as real marginal costs are
higher)! the Central Bank will react to inflation by increasing the
nominal interest rate more than proportionally ! as a result the
real interest rate will increase (lean against the wind policy).
As a result y will increase less than proportionally.
Remark: no comovement between c and g .
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
33 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Government Spending Shock with a weak response to deviations from
targets
ιπ = 1.1; ιy = 0
B. Annicchiarico
Output, Y
C ons umption, c
0
Labour, L
0.1
0.14
0.08
0.12
-0.02
Inv es tments , I
0.1
0
0.1
0.06
-0.1
%
%
0.08
-0.04
0.04
0.06
0.02
0
-0.08
0
20
-0.3
0.02
-0.02
40
0
20
quarters
40
0
0
20
40
-0.4
0
quarters
Inflation
R eal W age, W /P
0.03
0
3.5
-3
R
x eal
10 Interes t R ate
3
-0.01
40
Public D ebt, B/P
1
0.8
2.5
0.02
0.015
20
quarters
0.025
2
0.6
1.5
0.4
-0.02
0.01
1
-0.03
0.005
0
-0.2
0.04
-0.06
0.2
0.5
0
20
quarters
40
-0.04
0
20
quarters
40
0
0
20
40
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
0
0
20
40
quarters
21 Giugno 2010
34 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Monetary Policy Shock
B. Annicchiarico
Consumption, c
Output, Y
0
Labour, L
0
Investments, I
0.5
-0.1
2
0
-0.5
-0.2
0
-0.5
%
-1
%
-0.3
-2
-1
-0.4
-0.5
0
20
40
-2
-6
-2
-1.5
-0.6
-0.7
-4
-1.5
0
20
quarters
40
-2.5
-8
-3
-10
0
20
40
0
quarters
Inflation
Real Wage, W/P
0.5
0
0
20
40
quarters
Real Interest Rate
Public Debt, B/P
0.045
1.4
-0.2
0.04
1.2
-0.4
0.035
1
-0.6
0.03
0.8
-0.8
0.025
0.6
-1
0.02
0.4
-1.2
0.015
0.2
-1.4
0.01
-0.5
-1
-1.5
0
20
quarters
40
0
20
quarters
40
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
35 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Monetary Policy Shock: Propagation
The presence of price rigidities is a source of nontrivial real effects of
monetary policy shocks.
Firms cannot immediatly adjust the price of their good when they
receive new information about costs or demand conditions.
The shock generates an increase in the real rate, a decrease in inflation,
output and employment.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
36 / 65
Model #1: A NK Model with Capital Accumulation
Effects of a Monetary Policy Shock with a higher degree of price rigidities
B. Annicchiarico
Consumption, c
Output, Y
0
Labour, L
0
Investments, I
2
-1
5
0
0
-0.5
-2
-5
-1
%
%
-2
-3
-10
-4
-4
-15
-1.5
-6
-5
-2
0
20
40
-6
0
20
quarters
40
-8
-20
0
20
40
-25
0
quarters
Inflation
Real Wage, W/P
0.5
20
40
quarters
Real Interest Rate
0
Public Debt, B/P
0.15
1.5
0.1
-0.5
0
1
0.05
-1
0
-0.5
-1.5
0.5
-0.05
-2
-0.1
-1
-2.5
-1.5
0
20
quarters
40
-3
0
-0.15
0
20
quarters
40
-0.2
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
-0.5
0
20
40
quarters
21 Giugno 2010
37 / 65
Model #2: Model #1+ Habit+Adjust. Costs
Extend the previous model to account for:
external habit (see Lecture I)
adjustment costs on investments (see Lecture I)
adjustment costs on investments on labour (see Lecture I)
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
38 / 65
Model #2: Model #1+ Habit+Adjust. Costs
Households and Preferences
The typical household will solve the following problem
Household i is characterized by the following lifetime utility function:
∞
E0
∑ βt
log(Ci ,t
he C t
1) +
t =0
ω
1
v
(1
Li ,t )1
v
C average aggregate consumption; he measure of habit intensity
The period-by-period budget constraint is
Pt Ci ,t + Bi ,t + Pt Ii ,t
= Wi ,t Li ,t + (1 + rt 1 )Bi ,t 1 + Pt rtK Ki ,t
+Di ,t Pt TAXt Pt ADJ (It , Kt )
where
B. Annicchiarico
Ki ,t +1 = (1
γ
ADJ (Ii ,t , Ki ,t ) = I
2
δ) Ki ,t + Ii ,t
Ii ,t
Ki ,t
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
2
δ
Ki ,t
21 Giugno 2010
39 / 65
Model #2: Model #1+ Habit+Adjust. Costs
Households
At the optimum we now have (dropping index i )
Ct
1
he C t
= λt
1
λt
λ t +1
= βEt
(1 + rt )
Pt
Pt +1
γI
It
Kt
δK
= qt
qt λt = βEt λt +1 rt +1 + β (1
|{z}
ξt
βEt λt +1
1
δ) Et qt +1 λt +1
| {z }
ξ t +1
∂ADJ (It +1 , Kt +1 )
∂Kt +1
where qt is the Tobin’s marginal q.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
40 / 65
Model #2: Model #1+ Habit+Adjust. Costs
Intermediate-goods firms the adjustment costs on labour
We now assume that hiring and firing unionized workers is costly, in
particular we have:
ADJ_Lt (j ) =
γL
2
Lt (j )
Lt 1 (j )
2
1
Yt
As a result the optimal demand of labor is
B. Annicchiarico
Wt
Pt
= αL MCt (J )
+ βγL Et
Yt (j )
Lt (j )
λRt+1
λRt
γL
Lt +1 (j )
Lt (j )
Lt (j )
Lt 1 (j )
1
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
1
Lt +1 (j )
Lt (j )2
Yt
Lt
1
(j )
+
Yt +1
21 Giugno 2010
41 / 65
Model #2: Model #1+ Habit+Adjust. Costs
The resource constrain of the economy
The resource constraint of the economy is now
B. Annicchiarico
Yt
= Ct + It + Gt +
γp
+ (Πt 1)2 Yt +
2
2
γL
Lt (j )
+
1 Yt +
2 Lt 1 (j )
+
γI
2
It
Kt
2
δ
Kt
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
42 / 65
Model #2: Model #1+ Habit+Adjust. Costs
Effects of a Technology Shock
B. Annicchiarico
Consumption, c
Output, Y
0.8
Labour, L
1.4
Investments, I
0.6
6
1.2
0.6
5
0.4
1
4
3
0.4
%
%
0.8
0.2
0.6
2
0.4
0.2
1
0
0.2
0
0
20
40
0
0
0
20
quarters
40
-0.2
0
20
40
0
quarters
Inflation
Real Interest Rate
1
0.1
40
Public Debt, B/P
0.2
0.1
0.8
-0.1
20
quarters
Real Wage, W /P
0
0.05
0
0.6
-0.1
-0.2
0
-0.2
0.4
-0.3
-0.4
-1
0.2
0
20
quarters
40
0
-0.3
-0.05
-0.4
0
20
quarters
40
-0.1
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
-0.5
0
20
40
quarters
21 Giugno 2010
43 / 65
Model #2: Model #1+ Habit+Adjust. Costs
Effects of a Government Spending Shock
B. Annicchiarico
Consumption, c
Output, Y
0
Labour, L
0.08
-0.02
0.06
-0.04
Investments, I
0.12
0.2
0.1
0.1
0
0.08
-0.08
0.02
-0.1
%
0.04
%
-0.06
0.06
-0.2
0.04
-0.1
0
-0.12
-0.14
0
20
-0.3
0.02
-0.02
40
0
20
quarters
40
-0.4
0
0
20
40
-0.5
0
quarters
Inflation
Real Wage, W /P
0.06
20
40
quarters
0
Real Interest Rate
0.03
0.05
0.025
0.8
-0.05
0.04
Public Debt, B/P
1
0.02
0.6
0.03
-0.1
0.015
0.4
0.02
0.01
-0.15
0.01
0
0.2
0.005
0
20
quarters
40
-0.2
0
20
quarters
40
0
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
44 / 65
Model #2: Model #1+ Habit+Adjust. Costs
Effects of a Monetary Policy Shock
B. Annicchiarico
Consumption, c
Output, Y
0
Labour, L
0
Investments, I
0.5
-0.1
2
0
-0.5
-0.2
0
-0.5
%
-1
%
-0.3
-2
-1
-0.4
-0.5
0
20
40
-2
-6
-2
-1.5
-0.6
-0.7
-4
-1.5
0
quarters
20
40
-2.5
-8
-3
-10
0
20
40
0
quarters
Inflation
Real Wage, W /P
0.5
0
Real Interest Rate
0.1
-0.2
0
20
40
quarters
0.08
1.5
-0.4
-0.5
-0.6
0.06
-1
-0.8
0.04
Public Debt, B/P
2
1
-1
-1.5
-2
0
20
quarters
40
-1.4
0.5
0.02
-1.2
0
20
quarters
40
0
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
45 / 65
Model #3: Model #1+RoT
Extend the Model #1 to account for:
Consider rule of thumb households as in Galí, López-Salido and
Vallés (2007).
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
46 / 65
Model #3: Model #1+RoT
Households
There is a continuum of households. Population is constant and
normalized to 1.
A fraction sNR of households do not borrow and save, and just
consume their current labor income (hand-to-mouth households) !
extreme form of non-Ricardian behavior
Motivation: an extensive empirical literature provides evidence of
“excessive” dependence of consumption on current income; deviations
from the permanent income hypothesis.
As a result now we have two types of households:
Non-Ricardian households (population share sNR )
Ricardian households (population share 1
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
sNR )
21 Giugno 2010
47 / 65
Model #3: Model #1+RoT
The Ricardian Households
The typical Ricardian household will solve the following problem
Household i is characterized by the following lifetime utility function:
∞
E0
∑ βt
log(CiR,t
he C t
t =0
1) +
ω
1
v
1
LRi,t
1 v
C average aggregate consumption; he measure of habit intensity
The period-by-period budget constraint is
Pt CiR,t + Bir,t + Pt Iir,t
= Wi ,t LRi,t + (1 + rt
+Pt rtK KiR,t
+ DiR,t
R
1 )Bi ,t 1
+
Pt TAXtR
where
B. Annicchiarico
KiR,t +1 = (1
δ) KiR,t + IiR,t
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
48 / 65
Model #3: Model #1+RoT
The Ricardian Households
At the optimum we now have (dropping index i) we have the
standard optimality conditions:
B. Annicchiarico
1
= λRt
CtR
λi ,t
λR
λRt
= βEt t +1 (1 + rt )
Pt
Pt +1
h
i
= βEt λi ,t +1 rtK+1 + (1 δ)
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
49 / 65
Model #3: Model #1+RoT
The Non-Ricardian Households
Non-Ricardian households are assumed to behave in a
“hand-to-mouth” fashion: they fully consume their current labor
income (no consumption smoothing).
The representative household of this category derives utility from
consumption and leisure:
log CiNR
,t
ω
1
v
1
LNR
i ,t
1 vNA
given a flow budget constraint of the form:
NR
Pt CiNR
,t = Wi ,t Li ,t
Pt TAXiNR
,t
Consumption function is then:
B. Annicchiarico
CiNR
,t =
Wi ,t NR
L
Pt i ,t
TAXiNR
,t
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
50 / 65
Model #3: Model #1+RoT
Aggregate variables
Aggregate consumption is now defined as
Ct
sNR CtNR + (1
sNR )CtR
while investments and capital aggregates of the economy are give by
B. Annicchiarico
It
(1
sNR )ItR
Kt
(1
sNR )KtR
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
51 / 65
Model #3: Model #1+RoT
Wage Setting
New assumption: The fraction of Non-Ricardian and Ricardian
households is uniformly distributed across workers types and hence
across unions. Each period a typical union representing worker i sets
the wage for its workers in order to maximize the objective function of
the form
sNR
+(1
ω
1
v
sNR )
(1
Li ,t )1
ω
1
v
(1
v
+ λNR
i ,t
Li ,t )1
W
Wi ,t
Li ,t +
Pt
Wi ,t
v
Li ,t
+ λRi,t
Pt
σL
s.t. to the demand schedule: Li ,t = Wi t,t
Lt . At the optimum the
wage equation is
h
i
σL
R Wt
ω (1 Lt ) v = sNR λNR
+
(
1
s
)
λ
NR
t
t
σL 1
Pt
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
52 / 65
Model #3: Model #1+RoT
The government
The government budget constraint is
Bt = (1 + rt )Bt
1
+ Pt Gt
Pt TAXt
where Gt = G exp(εg ,t ) and εg ,t = ρg εg ,t + ξ g ,t with ξ g ,t
while
TAXt
sNR TAXtNR + (1 sNR )TAXtR
iid.N (0, σ2g ),
where we assume that TAXtNR = TAXtR = TAXt
The fiscal rule is as before:
Pt TAXt = Pt TAX + τPt Bt
Set sNR = 0.20.
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
53 / 65
Model #3: Model #1+ RoT
Effects of a Technology Shock
B. Annicchiarico
Consumption, c
Output, Y
0.6
Labour, L
1.4
Investments, I
0.6
6
1.2
5
0.4
1
0.8
0.3
0.6
4
3
%
0.4
%
0.5
0.2
2
0.4
0.2
0.1
1
0
0.2
0
20
40
0
0
0
20
quarters
40
-0.2
0
20
40
-1
0
quarters
Inflation
Real Wage, W/P
0
Real Interest Rate
0.8
-0.05
20
40
quarters
0.05
0.6
-0.1
Public Debt, B/P
0.25
0.04
0.2
0.03
0.15
0.02
0.1
0.01
0.05
0.4
-0.15
-0.25
0
0.2
-0.2
0
-0.01
0
20
quarters
40
0
0
20
quarters
40
-0.02
-0.05
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
-0.1
0
20
40
quarters
21 Giugno 2010
54 / 65
Model #3: Model #1+ RoT
Effects of a Technology Shock: Consumption paths
B. Annicchiarico
1.8
C
CNR
CR
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
30
35
40
21 Giugno 2010
55 / 65
Model #3: Model #1+ RoT
Effects of a Government Spending Shock
B. Annicchiarico
Consumption, c
Output, Y
0
Labour, L
0.08
0.1
0
0.1
0.06
-0.02
Investments, I
0.12
-0.1
0.08
-0.2
%
%
0.04
-0.04
0.06
-0.3
0.02
0.04
-0.06
-0.08
0
0
20
40
-0.02
0
quarters
20
40
0
-0.5
0
20
40
-0.6
0
quarters
-3
6
-0.4
0.02
x 10 Inflation
Real Wage, W/P
-0.005
3
-0.01
5
20
40
quarters
-3
Real
x 10 Interest Rate
Public Debt, B/P
1
2.5
0.8
-0.015
4
-0.02
2
0.6
3
-0.025
1.5
0.4
1
0.2
-0.03
2
1
-0.035
0
20
quarters
40
-0.04
0
20
quarters
40
0.5
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
56 / 65
Model #3: Model #1+ RoT
Effects of a Government Spending Shock: Consumption paths
B. Annicchiarico
0.06
C
CNR
CR
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
5
10
15
20
25
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
30
35
40
21 Giugno 2010
57 / 65
Model #3: Model #1+ RoT
Effects of a Monetary Policy Shock
B. Annicchiarico
Consumption, c
Output, Y
-0.1
Labour, L
0
Investments, I
0.5
-0.2
2
0
-0.5
-0.3
0
-0.5
%
-1
%
-0.4
-2
-1
-0.5
-0.6
0
20
40
-2
-6
-2
-1.5
-0.7
-0.8
0
quarters
20
40
-2.5
-8
-3
-10
0
20
40
0
quarters
Inflation
0
0
20
40
quarters
Real Wage, W/P
0.5
Real Interest Rate
0.1
Public Debt, B/P
2
0.08
-0.5
-0.5
1.5
0.06
-1
1
-1
0.04
-1.5
-1.5
-2
-4
-1.5
0
20
quarters
40
-2
0.5
0.02
0
20
quarters
40
0
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
58 / 65
Model #3: Model #1+ RoT
Effects of a Monetary Policy Shock: Consumption Paths
B. Annicchiarico
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
C
CNR
CR
-4.5
-5
0
5
10
15
20
25
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
30
35
40
21 Giugno 2010
59 / 65
Model #4: Habit+Adjust. Costs+RoT
Effects of a Technology Shock
B. Annicchiarico
Consumption, c
Output, Y
0.6
1.2
0.5
1
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
Labour, L
Investments, I
0.6
6
5
0.4
4
3
%
1.4
%
0.7
0.2
0
0
20
40
0
2
1
0
0
0
20
quarters
40
-0.2
0
Inflation
Real Wage, W /P
0.04
0.1
1.2
1
-0.15
0.8
0.02
-0.2
0.6
0
-0.25
0.4
-0.3
0.2
40
0
20
quarters
40
0
-0.1
-0.02
0
20
40
Public Debt, B/P
0.2
-0.1
20
0
Real Interest Rate
0.06
-0.05
quarters
-1
quarters
1.4
0
40
quarters
0
-0.35
20
-0.04
-0.2
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
-0.3
0
20
40
quarters
21 Giugno 2010
60 / 65
Model #4: Habit+Adjust. Costs+RoT
Effects of a Government Spending Shock
B. Annicchiarico
Consumption, c
Output, Y
0.05
0.08
0
0.06
Labour, L
Investments, I
0.12
0.1
0.1
0
0.08
-0.1
0.02
-0.1
%
0.04
%
-0.05
0.06
-0.2
0.04
-0.15
-0.2
0
0
20
40
-0.02
-0.3
0.02
0
quarters
20
40
0
0
20
40
Inflation
0
0
Real Interest Rate
0.02
Public Debt, B/P
0.8
0.015
-0.1
0.02
40
1
-0.05
0.025
20
quarters
Real Wage, W /P
0.03
0.6
-0.15
0.015
0.01
-0.2
0.01
0.4
-0.25
0.005
0
-0.4
quarters
0.005
0.2
-0.3
0
20
quarters
40
-0.35
0
20
quarters
40
0
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
61 / 65
Model #4: Habit+Adjust. Costs+RoT
Effects of a Monetary Policy Shock
B. Annicchiarico
Consumption, c
Output, Y
0
Labour, L
0.5
-0.2
Investments, I
0.5
2
0
0
-0.4
0
-0.5
-1
-0.8
-1
-1.5
%
-0.5
%
-0.6
-2
-1
-1.2
-1.4
-6
-2
-1.5
0
20
40
-2
0
quarters
20
40
-4
-2.5
-8
-3
-10
0
20
40
0
quarters
Inflation
Real Wage, W /P
0.5
0
-0.5
0
20
40
quarters
Real Interest Rate
Public Debt, B/P
0.25
2.5
0.2
2
0.15
1.5
0.1
1
0.05
0.5
-1
-0.5
-1.5
-1
-2
-1.5
-2
-2.5
0
20
quarters
40
-3
0
20
quarters
40
0
0
20
quarters
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
40
0
0
20
40
quarters
21 Giugno 2010
62 / 65
Discussion
What is still missing to have a more complete model?
Indexation
Wage rigidities
Variable capacity utilization
International trade and international capital markets
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
63 / 65
Discussion
Challenges of DSGE modelling
Labour migration and remittances
Foreign direct investments
Heterogenous workers (atypical, self-employed etc...)
Informal sector
Role of relative price movements
Non-market sector (public goods)
Financial market frictions
Portfolio choice
Term structure of interest rates
Currency risk premia
Endogenous growth
Time varying parameters and structural breaks
Estimation problems
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
64 / 65
References
Galí, J., López-Salido, J.D., Vallés, J., (2007), Understanding the Effects of Government
Spending on Consumption, Journal of the European Economic Association, 5(1).
Rotemberg, J., (1983), Aggregate Consequences of Fixed Costs of Price Adjustment,
American Economic Review, 73(3).
Ascari, G., Merkl, C., (2009), Real Wage Rigidities and the Cost of Disinflations,
Journal of Money, Credit and Banking, 41(2-3).
Ascari, G., Ropele, T., (2007), Optimal monetary policy under low trend inflation,
Journal of Monetary Economics, 54(8).
Ascari, G., Rossi, L. (2009), Real Wage Rigidities and Disinflation Dynamics: Calvo vs.
Rotemberg Pricing, University of Pavia.
Ascari, G., Rossi, L. (2010), Trend Inflation, Non-linearities and Firms Price-Setting:
Rotemberg vs. Calvo, University of Pavia.
Leeper, E. M., (1991), Equilibria under ’active’ and ’passive’ monetary and fiscal
policies, Journal of Monetary Economics 27(1).
B. Annicchiarico
(Università di Tor Vergata) (Institute)
Microfoundations of DSGE Models
21 Giugno 2010
65 / 65