Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
3
36
The Standard Real Business Cycle (RBC) Model
•
Perfectly competitive economy
•
Optimal growth model + Labor decisions
•
2 types of agents
– Households
– Firms
•
Shocks to productivity
•
Pareto optimal economy
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
Can be solved using a Social Planner program or solving for
a competitive equilibrium
•
37
We will solve for the equilibrium
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
3.1
38
The Household
•
Mass of agents = 1 (no population growth)
•
Identical agents + All face the same aggregate shocks (no
idiosyncratic uncertainty)
•;
Representative agents
39
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
Infinitely lived rational agent with intertemporal utility
Et
X
β sUt+s
s=0
β ∈ (0, 1):
•
discount factor,
Preferences over
– a consumption bundle
– leisure
• ; Ut = U (Ct, `t)
with U (·, ·)
– class C 2, strictly increasing, concave and satisfy Inada con-
40
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
ditions
– compatible with balanced growth [more below]:
(
U (Ct, `t) =
Ct1−σ
1−σ v (`t)
log(Ct) + v(`t)
if σ ∈ R+\{1}
if σ = 1
41
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Preferences are therefore given by


∞
X
Et 
β sU (Ct+s, `t+s)
s=0
•
Household faces two constraints
•
Time constraint
ht+s + `t+s 6 T
(for convenience T=1)
=1
42
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
Budget constraint
t+s }
| B{z
Bond purchases
+ C
+ It+s}
| t+s {z
Good purchases
6 (1 + rt+s−1)Bt+s−1 + W
{zht+s} +
|
{z
} | t+s
Bond revenus
•
W ages
z|t+s{z
Kt+s}
Capital revenus
Capital Accumulation
Kt+s+1 = It+s + (1 − δ )Kt+s
δ ∈ (0, 1):
Depreciation rate
43
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
The household decides on consumption, labor, leisure, investment, bond holdings and capital formation maximizing utility
constraint, taking the constraints into account
max
{Ct+s,ht+s,`t+s,It+s,Kt+s+1,Bt+s}∞
t=0


∞
X
β sU (Ct+s, `t+s)
Et 
s=0
subject to the sequence of constraints


ht+s + `t+s 6 1



B
t+s + Ct+s + It+s 6 (1 + rt+s−1)Bt+s−1 + Wt+sht+s + zt+sKt+s

Kt+s+1 = It+s + (1 − δ )Kt+s



 Kt, Bt−1 given
44
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
max
{Ct+s,ht+s,Kt+1,Bt+s}∞
t=0


∞
X
Et 
β sU (Ct+s, 1 − ht+s)
s=0
subject to
Bt+s+Ct+s+Kt+s+1 6 (1+rt+s−1)Bt+s−1+Wt+sht+s+(zt+s+1−δ )Kt+s
Write the Lagrangian

Lt = Et
P∞
sU (C
β
t+s, 1 − ht+s) + Λt+s
s=0
(1 + rt+s−1)Bt+s−1 +
!
+Wt+sht+s + (zt+s + 1 − δ )Kt+s − Ct+s − Bt+s − Kt+s+1 
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
First order conditions (∀ s ≥ 0)
Ct+s
ht+s
Bt+s
Kt+s+1
:
:
:
:
EtUc(Ct+s, 1 − ht+s) = EtΛt+s
EtU`(Ct+s, 1 − ht+s) = Et(Λt+sWt+s)
EtΛt+s = βEt((1 + rt+s)Λt+s+1)
EtΛt+s = βEt(Λt+s+1(zt+s+1 + 1 − δ ))
and the transversality condition
lim β sEtΛt+s(Bt+s + Kt+s+1) = 0
s−→+∞
45
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
46
Rearranging terms:
ht+s
Bt+s
Kt+s+1
: EtU`(Ct+s, 1 − ht+s) = EtUc(Ct+s, 1 − ht+s)Wt+s
: EtUc(Ct+s, 1 − ht+s) = βEt((1 + rt+s)EtUc(Ct+s+1, 1 − ht+s+1)
: EtUc(Ct+s, 1 − ht+s) = βEt(Uc(Ct+s+1, 1 − ht+s+1)(zt+s+1 + 1 −
and the transversality condition
lim β sEtUc(Ct+s, 1 − ht+s)(Bt+s + Kt+s+1) = 0
s−→+∞
47
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Simple example : Assume U (Ct, `t) = log(Ct) + θ log(1 − ht)
ht+s
Bt+s
Kt+s+1
t+s
: Et 1−h1 t+s = Et W
Ct+s
1
: Et C1t+s = βEt(1 + rt+s) Ct+s+1
1 (z
: Et C1t+s = βEt Ct+s+1
t+s+1 + 1 − δ )
and the transversality condition
Kt+s+1 + Bt+s
s
lim Etβ
s−→+∞
Ct+s
=0
48
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Remark :
- It is convenient to write and interpret FOC for s = 0:
ht
Bt
Kt+1
: U`(Ct, 1 − ht) = EtUc(Ct, 1 − ht)Wt
: Uc(Ct, 1 − ht) = βEt((1 + rt)EtUc(Ct+1, 1 − ht+1))
: Uc(Ct, 1 − ht) = βEt(Uc(Ct+1, 1 − ht+1)(zt+1 + 1 − δ ))
and the transversality condition
Kt+s+1 + Bt+s
s
lim Etβ
s−→+∞
Ct+s
=0
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
We have consumption smoothing and
•
We have labor smoothing
θ
Wt(1 − ht)
3.2
49
θ
= β (1 + rt)Et
Wt+1(1 − ht+1)
The Firm
•
Mass of firms = 1
•
Identical firms + All face the same aggregate shocks (no idiosyncratic uncertainty)
;
Representative firm
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
50
•
Produce an homogenous good that is consumed or invested
•
by means of capital and labor
•
Constant returns to scale technology (important)
Yt = AtF (Kt, Γtht)
•
Γt = γ Γt−1 Harrod neutral technological progress (γ > 1), At
stationary (does not explain growth)
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
51
- Remark: one could introduce long run technical progress in
three different ways:
b F (Γ
e K ,Γ h )
Yt = Γ
t
t t t t
b is Hicks Neutral, Γ
e is Solow neutral
-Γ
t
t
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
52
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
53
- Harrod neutral technical progress and the preferences specified above are needed for the existence of a Balanced Growth
Path that replicates Kaldor Stylized Facts:
1. The shares of national income received by labor and capital are roughly constant over long periods of
time
2. The rate of growth of the capital stock is roughly constant over long periods of time
3. The rate of growth of output per worker is roughly constant over long periods of time
4. The capital/output ratio is roughly constant over long periods of time
5. The rate of return on investment is roughly constant over long periods of time
6. The real wage grows over time
- End of the remark
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
54
Yt = AtF (Kt, Γtht)
•
Γt = γ Γt−1 Harrod neutral technological progress (γ > 1), At
stationary (does not explain growth)
• At
are shocks to technology. AR(1) exogenous process
log(At) = ρ log(At−1) + (1 − ρ) log(A) + εt
with εt ; N (0, σ2).
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
The firm decides on production plan maximizing profits
max AtF (Kt, Γtht) − Wtht − ztKt
{Kt,ht}
First order conditions:
Kt
ht
: AtFK (Kt, Γtht) = zt
: AtFh(Kt, Γtht) = Wt
Simple Example: Cobb–Douglas production function
Yt = AtKtα(Γtht)1−α
First order conditions
Kt
ht
: αYt/Kt = zt
: (1 − α)Yt/ht = Wt
55
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
3.3
56
Equilibrium
The (RBC) Model Equilibrium is given by the following equations
(∀ t ≥ 0):
1. Exogenous Processes : log(At) = ρ log(At−1)+(1−ρ) log(A)+εt
and Γt = γ Γt−1
2. Law of motion of Capital : Kt+1 = It + (1 − δ )Kt
3. Bond market equilibrium : Bt = 0
4. Good Markets equilibrium : Yt = Ct + It
57
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
5. Labor market equilibrium :
U`(Ct,1−ht)
Uc(Ct,`t)
= AtFh(Kt, Γtht)
6. Consumption/saving decision + Capital market equilibrium
:
Uc (Ct , 1 − ht ) = βEt [Uc (Ct+1 , 1 − ht+1 )(At+1 FK (Kt+1 , Γt+1 ht+1 ) + 1 − δ)]
7. Financial markets :
1 + rt =
Et [Uc (Ct+1 , 1 − ht+1 )(At+1 FK (Kt+1 , Γt+1 ht+1 ) + 1 − δ)]
Et Uc (Ct+1 , 1 − ht+1 )
58
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
3.4
An Analytical Example
• U (Ct, `t) = log(Ct) + θ log(`t),
•
Yt = AtKtα(Γtht)1−α
Equilibrium
θCt
1 − ht
Yt
= (1 − α)
ht
1
1
Yt+1
= βEt
+1−δ
Ct
Ct+1 Kt+1
Kt+1 = Yt − Ct + (1 − δ )Kt
Yt = AtKtα(Γtht)1−α
Kt+1+s
s
=0
lim β Et
s−→∞
Ct+s
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
3.5
Stationarization
•
We want a stationary equilibrium (technical reasons)
•
Deflate the model for the growth component Γt
•
On the example: xt = Xt/Γt
59
60
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
- Deflated Equilibrium
θct
yt
= (1 − α)
1 − ht ht
1 yt+1
1 β
+1−δ
= Et
ct
γ
ct+1 kt+1
γkt+1 = yt − ct + (1 − δ )kt
yt = Atktαh1−α
t
lim
s−→∞
γkt+1+s
s
β
ct+s
=0
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
4
61
Solving the Model
4.1
•
In General
Non–linear system of stochastic finite difference equations under rational expectations
•
Very complicated
•
In general no analytical solution, need to rely on numerical
approximation methods
62
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
4.2
The Nice Analytical Case
• U (c, `) = log(c) + θ log(`), γ
•
= 1 (not needed) and δ = 1
Equilibrium
θct
1 − ht
yt
= (1 − α)
ht 1 yt+1
1
= βEt
ct
ct+1 kt+1
kt+1 = yt − ct
yt = Atktαh1−α
t
lim
s−→∞
γkt+1+s
s
β
ct+s
=0
63
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
The solution is: ct = (1 − αβ )yt
1−α
⇒ kt+1 = αβκAtktα
• ht = h = 1−α+θ(1−αβ)
with κ = h1−α
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
4.3
64
Numerical solution
•
Unfortunately: No closed form solution in general
•
Have to adopt a numerical approach
•
Log–linearize the equilibrium around the steady state (limits?)
•
Solve the linearized model using standard techniques
65
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
The solution to the log–linearize version of the model takes
the form
Xt+1
= MxXt + Mz Zt
(1)
Zt+1
= ρZt + εt+1
(2)
= PxXt + Pz Zt
(3)
Yt
where Xt, Zt and Yt collect, respectively, the state variables,
the shocks and the variables of interest
•
Resembles a VAR model
66
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
In the basic RBC model
Xt =
{kt}
(4)
Zt =
{at}
(5)
Yt = {yt, ct, it, ht}
•
(6)
Let’s evaluate the quantitative ability of the model to account for and explain the cycle.
67
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
ct 6
∆ct = 0
o
?
/
6
/
}
i
/
I
I
c⋆
R
R
c0
6
-
7
•
R
R
-
^
∆kt = 0
?
-
k0
k⋆
kt
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
5
Quantitative Evaluation
5.1
Calibration
•
Need to assign numerical values to the parameters
•
This is the calibration step
•
What does calibration mean?
– Make explicit use of the model to set the parameters
– A lot of discipline, but no systematic recipe
– Have to set (α, β, θ, δ, γ, ρ, σ, A)
68
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
– Use data: (k/y, c/y, i/y, h, wh/y, r)
– Compute (k/y, c/y, i/y, h, wh/y, r) in the model
69
70
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
In the data ∆y = 0.9% per quarter ; γ = 1.009.
•
In the data i/k = 0.076 on annual data. Use capital accumulation to get annual depreciation.
i/k (model) = gamma − (1 − δ ) ; δ
= 0.01
•
In the data wh/y = 0.6, in the model wh/y = 1 − α ; α = 0.4.
•
In the data k/y = 3.32 on annual data. Using the Euler
equation
β
Then β = 0.98
=
γ
αy/k + (1 − δ )
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
• h = 0.31,
such that θ = (1−α)(1−h)
hc/y
71
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
5.2
72
Can the Business Cycle be Driven by Capital Dynamics?
•
Want to see whether capital dynamics can account for BC.
•
Perfect foresights dynamics
•
We study a case with fixed labor (h = h) and a variable labor
case (U (c, 1 − h))
73
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Capital shock - Fixed hours
Output
Consumption
0
−0.3
−0.6
0.5
−0.8
40 60
Quarters
−1
80
20
40 60
Quarters
Hours worked
0.4
−0.2
0.3
−0.4
0.2
% dev.
0
−0.6
−0.8
40 60
Quarters
80
40 60
Quarters
80
−0.1
−0.2
0.1
−0.1
20
Labor productivity (Wages)
0
−0.3
−0.4
0
20
0
80
% dev.
20
Capital
% dev.
1
−0.4
% dev.
% dev.
% dev.
−0.2
−1
1.5
−0.2
−0.1
−0.4
Investment
0
−0.5
20
40 60
Quarters
80
20
40 60
Quarters
80
74
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Capital shock - Variable hours
Output
Consumption
0
−0.3
−0.6
0.5
−0.8
40 60
Quarters
−1
80
20
40 60
Quarters
Hours worked
0.4
−0.2
0.3
−0.4
0.2
% dev.
0
−0.6
−0.8
40 60
Quarters
80
40 60
Quarters
80
−0.1
−0.2
0.1
−0.1
20
Labor productivity (Wages)
0
−0.3
−0.4
0
20
0
80
% dev.
20
Capital
% dev.
1
−0.4
% dev.
% dev.
% dev.
−0.2
−1
1.5
−0.2
−0.1
−0.4
Investment
0
−0.5
20
40 60
Quarters
80
20
40 60
Quarters
80
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
75
The answer is clearly that capital dynamics cannot be the
story for the business cycle.
•
Solow (1957): capital accumulation accounts for 1/8th of output growth.
•
Technical progress, not capital accumulation, is the engine of
growth.
•
At the business frequency: transitional dynamics does not
conform to the data (c and i for ex).
•
More (shocks?) is needed to understand the BC
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
76
That’s why the RBC literature proposes technological shocks
(Brock & Mirman, 1972)
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
5.3
77
Technological Shocks
•
How to calibrate the shocks?
•
We have
log(At) = log(yt) − α log(kt) − (1 − α) log(ht)
•
How to get kt? Use Capital accumulation with k0 = (k/y )y0
•
Estimate
log(At) = ρ log(At−1) + εt
We get ρ = 0.95 and σ = 0.0079. We set A = 1 without loss
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
of generality.
78
79
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
A good fit with estimated shocks
Output
Hours Worked
0.1
0.04
Data
Model
0.05
0.02
0
0
−0.02
−0.05
−0.04
−0.1
1950 1960 1970 1980 1990 2000
Quarters
−0.06
1950 1960 1970 1980 1990 2000
Quarters
Consumption
Investment
0.03
0.3
0.02
0.2
0.01
0.1
0
−0.01
0
−0.02
−0.1
−0.03
1950 1960 1970 1980 1990 2000
Quarters
−0.2
1950 1960 1970 1980 1990 2000
Quarters
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
80
A first success?
•
Accounts for the main events in the data
•
The model correctly predicts the data: corr(y, ym)=0.75,
corr(c, cm)=0.73, corr(i, im)=0.70.
•
BUT: corr(h, hm)=0.06
•
Let’s compute unconditional moments in the model (simulateHP filter-compute moments)
81
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
5.4
Results from Model Simulations
Variable
Output
σ (·) σ (·)/σ (y ) ρ(·, y ) ρ(·, h)
1.70
1.49
Consumption
0.80
0.37
Investment
6.49
5.00
Hours worked
1.69
0.85
Labor productivity 0.90
0.67
(model in yellow)
–
–
0.47
0.25
3.83
3.35
1.00
0.57
0.53
0.45
Auto
–
– 0.84
–
– 0.68
0.78
– 0.83
0.81
– 0.82
0.84
– 0.81
0.99
– 0.68
0.86
– 0.89
0.98
– 0.68
0.41 0.09 0.69
0.97 0.92 0.72
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
5.5
•
82
A success(?)
The model correctly predicts the amplitude, serial correlation
and relative variability of fluctuations
•
It accounts for a large part of output volatility
•
Correct ranking of the volatility of c, i, y, . . .
•
Large serial correlation, although it is smaller than in the
data.
•
But
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
– C and N are not volatile enough
– w (and r) are to too procyclical
83
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
6
84
Criticisms
6.1
•
In General
The research on RBC became so successful because
– Propose a coherent platform to analyse growth and cycles
– It somewhat fails such that there is room for work
•
Main victory: methodological (part of the toolkit of macroeconomists)
•
Main criticisms
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
85
– incorrect/implausible calibration of parameters (IES, Labor supply) ; need for sensibility analysis
– counterfactual prediction for some prices:
∗
real wage is strongly procyclical in the model,
∗
CRRA preferences are not compatible with the equity
premium
∗
6.2
•
price level is too strongly countercyclical
The Measure of Technological Shocks
Key problem: Are technological shocks at the source of BC?
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
86
Prescott [1986]: Technology shocks account for 70% of output
volatility, but
– Too volatile
– Little evidence of large supply shocks (except oil prices)
– Recessions have to be explained by technological regressions
– Measurement problems (contamination by demand shocks
if increasing returns, imperfect competition, labor hoarding) ; in the growth accounting literature, the SR was
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
87
a measure of our ignorance, now it is the engine of the
model.
6.3
•
The Need for Persistent Shocks
Recall that
log(At) = ρ log(At−1) + εt
and ρ is large
•
Why do we need so persistent shocks?
•
Because the model possesses weak propagation mechanisms
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
Let’s see that in details
88
89
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock
Technology shock
1
0.8
% dev.
0.6
0.4
0.2
0
10
20
30
40
Quarters
50
60
70
80
90
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock
Consumption
Investment
6
1.5
0.6
4
1
0.5
0
% dev.
0.8
% dev.
% dev.
Output
2
0.4
0.2
20
40
60
Quarters
80
0
Capital
0
20
40
60
Quarters
−2
80
Hours worked
0.8
2
20
40
60
Quarters
80
Labor productivity (Wages)
0.8
1
0.6
0.6
0.4
% dev.
% dev.
% dev.
0.5
0.4
0
0.2
0
0.2
20
40
60
Quarters
80
−0.5
20
40
60
Quarters
80
0
20
40
60
Quarters
80
91
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock: Temporary vs Persistent
Technology shock
1
0.8
% dev.
0.6
0.4
0.2
0
10
20
30
40
Quarters
50
60
70
80
92
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock: Temporary vs Persistent
Consumption
1.5
0.6
1
0.5
0.4
0.2
20
40
60
Quarters
80
0
0.6
1
0.4
0.2
20
40
60
Quarters
40
60
Quarters
80
20
40
60
Quarters
80
Labor productivity (Wages)
0.8
0.6
0.5
−0.5
2
−2
80
0
20
4
0
Hours worked
1.5
% dev.
% dev.
Capital
0.8
0
6
% dev.
0
Investment
8
% dev.
0.8
% dev.
% dev.
Output
2
0.4
0.2
20
40
60
Quarters
80
0
20
40
60
Quarters
80
93
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock: Persistent vs Permanent
Technology shock
1
0.8
% dev.
0.6
0.4
0.2
0
10
20
30
40
Quarters
50
60
70
80
94
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock: Persistent vs Permanent
Consumption
Investment
2
6
1.5
1.5
4
1
0.5
0
% dev.
2
% dev.
% dev.
Output
1
0.5
20
40
60
Quarters
80
0
Capital
0
20
40
60
Quarters
−2
80
Hours worked
2
2
20
40
60
Quarters
80
Labor productivity (Wages)
2
1
1.5
1.5
1
% dev.
% dev.
% dev.
0.5
1
0
0.5
0
0.5
20
40
60
Quarters
80
−0.5
20
40
60
Quarters
80
0
20
40
60
Quarters
80
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
How to improve the model:
– Introducing additional shocks
– Improving the propagation mechanisms
95
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
7
Solving the puzzles?
7.1
Adding Demand Shocks
•
Government Expenditures
•
Change the Household’s budget constraint:
Bt+1 + Ct + It + Tt 6 (1 + rt−1)Bt + Wtht + ztKt
•
Government Balanced Budget: Tt = Gt
•
Government expenditures are modeled as
log(Gt) = ρ log(Gt−1) + (1 − ρg ) log(G) + εg,t
96
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
with εg,t ; N (0, σt2).
• G/Y
= 0.2, ρg = 0.97, σg = 0.02.
97
98
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Government Spending Shock
Government shock
1
0.8
% dev.
0.6
0.4
0.2
0
10
20
30
40
Quarters
50
60
70
80
99
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Government Spending Shock
Output
Consumption
0.12
Investment
0
0.01
0.1
0
−0.05
0.06
% dev.
% dev.
% dev.
0.08
−0.1
0.04
−0.15
x 10
−3
40 60
Quarters
−0.2
80
Capital
−0.04
80
0.15
−4
−6
40 60
Quarters
80
0.1
0
40 60
Quarters
80
−0.02
0.05
20
20
Labor productivity (Wages)
0
0.2
% dev.
% dev.
40 60
Quarters
Hours worked
−2
−8
20
% dev.
0
20
−0.02
−0.03
0.02
0
−0.01
−0.04
−0.06
20
40 60
Quarters
80
−0.08
20
40 60
Quarters
80
100
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Technological and Government Spending Model
Variable
Output
σ (·) σ (·)/σ (y ) ρ(·, y ) ρ(·, h)
1.70
1.43
Consumption
0.80
0.62
Investment
6.49
4.62
Hours worked
1.69
0.85
Labor productivity 0.90
0.76
(model in yellow)
–
–
0.47
0.43
3.83
3.24
1.00
0.59
0.53
0.53
–
– 0.84
–
– 0.68
0.78
– 0.83
0.54
– 0.75
0.84
– 0.81
0.97
– 0.68
0.86
– 0.89
0.90
– 0.68
0.41 0.09 0.69
0.87 0.58 0.72
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
7.2
101
Labor indivisibility
•
Hansen [1985]: work a fixed amount of hours or does not
•
Preferences
(
U (Ct, 1 − ht) =
•
log(Cite ) + θ log(1 − h0) if she works
log(Citu ) + θ log(1)
if not
Randomly drawn with probability πt
Ut
= πit (log(Cite ) + log(1 − n0)) + (1 − πit) (log(Citu ) + log(1))
= πit log(Cite ) + (1 − πit) log(Citu ) − θht
where hit = πitn0
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
•
There exists full insurance
(
•
102
e + τ A + Ke
Cit
t it
it+1 6 (zt + 1 − δ )Kt + wth0
u + τ A + Ku
Cit
t it
it+1 6 (zt + 1 − δ )Kt + Ait
Risk neutral insurance companies: τt = πt
if she works
if not
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Labor indivisibility
•
Utility collapses to
U (Ct, 1 − ht) = log(Ct) − ht
•
the rest remains unchanged
103
104
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock - Indivisible Labor Model
Output
Consumption
% dev.
1
0.5
0
8
0.8
6
0.6
4
0.4
0.2
20
40 60
Quarters
0
80
20
40 60
Quarters
−2
80
Hours worked
2
0.8
1.5
0.8
0.6
1
0.6
0.4
0.2
0
0.5
0
20
40 60
Quarters
80
−0.5
20
40 60
Quarters
80
Labor productivity (Wages)
1
1
% dev.
% dev.
Capital
2
0
% dev.
% dev.
1.5
Investment
1
% dev.
2
0.4
0.2
20
40 60
Quarters
80
0
20
40 60
Quarters
80
105
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
IRF to A Technological Shock - Indivisible Labor Model
Output
Consumption
% dev.
1
0.5
0
8
0.8
6
0.6
4
0.4
0.2
20
40 60
Quarters
0
80
20
40 60
Quarters
−2
80
Hours worked
2
0.8
1.5
0.8
0.6
1
0.6
0.4
0.2
0
0.5
0
20
40 60
Quarters
80
−0.5
20
40 60
Quarters
80
Labor productivity (Wages)
1
1
% dev.
% dev.
Capital
2
0
% dev.
% dev.
1.5
Investment
1
% dev.
2
0.4
0.2
20
40 60
Quarters
80
0
20
40 60
Quarters
80
106
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
Indivisible Labor Model
Variable
Output
σ (·) σ (·)/σ (y ) ρ(·, y ) ρ(·, h)
1.70
1.95
Consumption
0.80
0.45
Investment
6.49
6.66
Hours worked
1.69
1.63
Labor productivity 0.90
0.45
(model in yellow)
–
–
0.47
0.23
3.83
3.40
1.00
0.83
0.53
0.23
–
– 0.84
–
– 0.68
0.78
– 0.83
0.78
– 0.83
0.84
– 0.81
0.99
– 0.67
0.86
– 0.89
0.98
– 0.67
0.41 0.09 0.69
0.77 0.65 0.83
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
107
Contents
1 Introduction
2
2 Measuring the Business Cycle
4
2.1 Trend versus Cycle . . . . . . . . . . . . . . . . . . .
4
2.1.1
Cycle: Output Gap . . . . . . . . . . . . . .
7
2.1.2
Growth Cycle . . . . . . . . . . . . . . . . . .
10
2.1.3
Trend Cycle . . . . . . . . . . . . . . . . . . .
12
2.1.4
The Hodrick–Prescott Filter . . . . . . . . .
14
2.1.5
The HP filter at work . . . . . . . . . . . . .
16
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
2.2 U.S. Business Cycles . . . . . . . . . . . . . . . . . .
108
19
2.2.1
What are Business Cycles? . . . . . . . . . .
19
2.2.2
Main Real Aggregates . . . . . . . . . . . . .
21
2.2.3
Moments . . . . . . . . . . . . . . . . . . . .
28
2.3 A Model to Replicate Those Facts . . . . . . . . . .
32
3 The Standard Real Business Cycle (RBC) Model 36
3.1 The Household . . . . . . . . . . . . . . . . . . . . .
38
3.2 The Firm . . . . . . . . . . . . . . . . . . . . . . . .
49
3.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . .
56
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
109
3.4 An Analytical Example . . . . . . . . . . . . . . . .
58
3.5 Stationarization . . . . . . . . . . . . . . . . . . . . .
59
4 Solving the Model
61
4.1 In General . . . . . . . . . . . . . . . . . . . . . . . .
61
4.2 The Nice Analytical Case . . . . . . . . . . . . . . .
62
4.3 Numerical solution . . . . . . . . . . . . . . . . . . .
64
5 Quantitative Evaluation
5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . .
68
68
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
110
5.2 Can the Business Cycle be Driven by Capital Dynamics? . . . . . . . . . . . . . . . . . . . . . . . . .
72
5.3 Technological Shocks . . . . . . . . . . . . . . . . . .
77
5.4 Results from Model Simulations . . . . . . . . . . .
81
5.5 A success(?) . . . . . . . . . . . . . . . . . . . . . . .
82
6 Criticisms
84
6.1 In General . . . . . . . . . . . . . . . . . . . . . . . .
84
6.2 The Measure of Technological Shocks . . . . . . . .
85
6.3 The Need for Persistent Shocks . . . . . . . . . . .
87
Franck Portier – TSE – Macro II – 2009-2010 – Chapter 3 – Real Business Cycles
7 Solving the puzzles?
7.1 Adding Demand Shocks . . . . . . . . . . . . . . . .
111
96
96
7.2 Labor indivisibility . . . . . . . . . . . . . . . . . . . 101