Problem set 6The RBC model
Markus Roth
Chair for MacroeconomicsJohannes Gutenberg Universität Mainz
February 11, 2011
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 1 / 23
Contents
1 Problem 1 (The Real Business Cycle Model)
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 2 / 23
Problem 1 (The Real Business Cycle Model)
Contents
1 Problem 1 (The Real Business Cycle Model)
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 3 / 23
Problem 1 (The Real Business Cycle Model)
The model
• We consider a representative household that maximizes
Et
∞
∑s=0
βs
[
lnCt+s + θ(1−Nt+s)1−γ
1− γ
]
(1)
subject to
Yt = (AtNt)α K1−α
t (2)
Kt+1 = (1− δ)Kt + Yt − Ct. (3)
• Like always we solve this problem using a Lagrangian function
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 4 / 23
Problem 1 (The Real Business Cycle Model)
The Lagrangian
• The Lagrangian is given by
L = Et
∞
∑s=0
βs
{
[
lnCt+s + θ(1−Nt+s)1−γ
1− γ
]
+ λt+s
[
(1− δ)Kt+s + (At+sNt+s)αK1−α
t+s − Ct+s − Kt+s+s
]
}
.
• The first two first order conditions are given by
∂L
∂Ct+s=
1
Ct+s− λt+s
!= 0
∂L
∂Kt+s+1= −λt+s + λt+s+1β
[
(1− δ) + (1− α)(At+sNt+s)αK−α
t+s
] != 0.
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Problem 1 (The Real Business Cycle Model)
The Euler equation
• Combining both optimality conditions yields the Euler equation
1
Ct= βEt
[
1− δ + (1− α)
(
At+1Nt+1
Kt+1
)α] 1
Ct+1, (4)
where we define the gross rate of return on capital by
Rt,t+1 ≡ 1− δ + (1− α)
(
At+1Nt+1
Kt+1
)α
. (5)
• The first order condition with respect to labor supply is
∂L
∂Nt+s= −θ(1−Nt+s)
−γ + λt+sαAαt+sN
α−1t+s K
1−αt+s
!= 0
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 6 / 23
Problem 1 (The Real Business Cycle Model)
Optimal labor supply
• Optimal labor supply is determined implicitly by
θ(1−Nt)−γ =
1
CtαAα
t
(
Kt
Nt
)1−α
. (6)
• Note that the model consisting of equations (2) to (6) togetherwith the technology shock process
lnAt = g+ φ lnAt−1 + εt (7)
is a nonlinear rational expectations model.
• In order to solve the model we use a log-linear approximation tothe system.
• Therefore we need the steady-state levels of the variables.
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Problem 1 (The Real Business Cycle Model)
Steady-state levels 1
• We start with the definition of the interest rate in the steady statefrom equation (5)
R = 1− δ + (1− α)
(
AN
K
)α
AN
K=
(
r+ δ
1− α
) 1α
, (8)
where we used r = R− 1.
• For the output/capital ratio we get from the production function(2)
Y
K=
(
AN
K
)α
=r+ δ
1− α. (9)
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Problem 1 (The Real Business Cycle Model)
Steady-state levels 2
• The investment/capital ratio is determined from the capitalaccumulation equation (3)
(1+ g)K = (1− δ)K+ I
1+ g = 1− δ +I
KI
K= g+ δ. (10)
• Note that since there is no population growth capital in the steadystate grows at rate (1+ g).
• From those results we can easily figure out C/K
C
K=
Y
K−
I
K=
r+ δ − (1− α)(g+ δ)
1− α. (11)
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Problem 1 (The Real Business Cycle Model)
Log-linearization 1
• We start with log-linearizing the production function.
• Recall that log-linearizing means that we express the model inlog-deviations from their steady-state.
• Due to the simple structure we can use a short-cut and divide theproduction function by its steady-state
Yt
Y=
(AtNt)αK1−αt
(AN)αK1−α.
• Then we take the natural logarithm on both sides, this yields
yt = α(at + nt) + (1− α)kt, (12)
where log-deviations of variables from their steady-state aredenoted by a hat (xt = ln(Xt/X)).
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Problem 1 (The Real Business Cycle Model)
Log-linearization 2
• Next we linearize the capital accumulation equation (3).
• We write the function as
(1+ g)Kekt+1 = (1− δ)Kekt + Yeyt − Cect .
• The left hand side is approximated by
LHS ≃ (1+ g)[
K+ K(kt+1 − k)]
.
• The right hand side is approximated as
RHS ≃ (1− δ)K+ Y− C+ (1− δ)K(kt − k) + Y(yt − y)− C(ct − c).
• Note that x = ln(X/X) = 0.
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Problem 1 (The Real Business Cycle Model)
Log-linearization 3
• Equating LHS and RHS yields
kt+1 =
[
(1− δ) +Y
K(1− α)
]
kt +Y
Kα(at + nt)−
C
Kct. (13)
• In order to linearize the Euler equation we write it as
EtCect+1 = CectβEtRe
rt+1.
• The left hand side is approximated by
LHS ≃ EtC(ct − c).
• The right hand side is approximated by
RHS ≃ CβR+ CβR(ct + rt+1).
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Problem 1 (The Real Business Cycle Model)
Log-linearization 4
• Equating both sides yields
Etct+1 − ct = Et∆ct+1 = rt+1. (14)
• The interest rate can be written as
Rert+1 = 1− δ + (1− α)
(
Aeat+1Nent+1
Kekt+1
)
.
• The left hand side is approximated as
LHS ≃ R+ Rrt+1.
• The right hand side is approximated as
RHS ≃ 1− δ+(1− α)
(
AN
K
)α
+ α(1− α)
(
AN
K
)α
(at+1+ nt+1− kt+1)
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Problem 1 (The Real Business Cycle Model)
Log-linearization 5
• Equating both sides yields
Rrt+1 = α(1− α)
(
AN
K
)α
(at+1 + nt+1 − kt+1)
(1+ r)rt+1 = α(1− α)r+ δ
1− α(at+1 + nt+1 − kt+1)
rt+1 =α(r+ δ)
1+ r(at+1 + nt+1 − kt+1). (15)
• The last equation we have to linearize is the labor supplycondition (6)
Cectθ(1−Nent)−γ = αAαeαatK1−αe(1−α)ktNα−1e(α−1)nt.
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Problem 1 (The Real Business Cycle Model)
Log-linearization 6
• The left hand side is approximated by
LHS ≃ Cθ(1−N)−γ + Cθ[(1−N)−γct − (1−N)−γ−1nt].
• The right hand side is approximated as
RHS ≃ αAαK1−αNα−1[1+ αat + (1− α)kt + (α − 1)nt]
• Equating the left hand side and the right hand side yields
ct +γN
N− 1nt = αat + (1− α)kt + (α − 1)nt.
• Rearranging yields
nt = αN− 1
γN+ 1− α−
N− 1
γN+ 1− αct + (1− α)
N− 1
γN+ 1− α. (16)
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 15 / 23
Problem 1 (The Real Business Cycle Model)
Matrix notation
• The log-linearized system consists of the equations (12), (13), (14)and (16).
• We write the system in matrix notation in the from of
AEtyt+1 = Byt +Cxt,
where
yt =
ytctntrtktat−1
.
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 16 / 23
Problem 1 (The Real Business Cycle Model)
Matrix A
• Matrix A is given by
0 0 0 0 0 α
0 1 −α(r+δ)1+r 0 α(r+δ)
1+r −ρ α(r+δ)1+r
0 0 0 0 0α(N−1)
γN
0 0 0 0 0α(r+δ)1+r
0 0 0 0 1 YKα
0 0 0 0 1 1
.
Markus Roth (Advanced Macroeconomics) Problem set 6 February 11, 2011 17 / 23
Problem 1 (The Real Business Cycle Model)
Matrix B
• Matrix B is given by
1 0 −α 0 α − 1 00 1 0 0 0 0
0 N−1γN+(N−1)(1−α)
1 0 (α − 1) N−1γN+(N−1)(1−α)
0
0 0α(r+δ)1+r 1 −
α(r+δ)1+r 0
0 −CK
YKα 0
[
(1− δ + YK(1−α)
)]
0
0 0 0 0 0 φ
.
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Problem 1 (The Real Business Cycle Model)
Matrix C
• Finally matrix C is a column vector only and is given by
000001
.
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Problem 1 (The Real Business Cycle Model)
0 5 10 15 200.7
0.75
0.8
0.85
0.9
0.95Output
0 5 10 15 200.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Labor
0 5 10 15 20−5
0
5
10
15Real Rate (BPs)
Figure: Impulse response functions of the RBC model.
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Problem 1 (The Real Business Cycle Model)
Impulse response functions
• Impulse response functions plot the response of the variables inthe system to a (usually) one percentage shock in the initialperiod.
• In our particular case this means that technology is shocked in theinitial period.
• Thereafter, it is assumed that there are no more shocks in theeconomy.
• The IRFs show how the variables react to the respective shock.
• For the basic RBC model derived above we find that hours, outputand the interest rate increase suddenly.
• Then they slowly approach their steady-state value again.
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Problem 1 (The Real Business Cycle Model)
Some comments
• Note that the specific representation of an RBC model we haveworked on is due to [Campbell, 1994].
• In his paper he considers different versions of the RBC model, welooked at the one with additive separable labor in the utilityfunction.
• Please note that we have not developed how the linearized systemis actually solved.
• For those who are interested in this issue should read[Blanchard and Kahn, 1980].
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References
References
Blanchard, O. J. and Kahn, C. M. (1980).The solution of linear difference models under rationalexpectations.Econometrica, 48(5):1305–11.
Campbell, J. Y. (1994).Inspecting the mechanism: An analytical approach to thestochastic growth model.Journal of Monetary Economics, 33(3):463–506.
Wickens, M. (2008).Macroeconomic Theory: A Dynamic General Equilibrium Approach.Princeton University Press.
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