Problem set 6 - Welcome to the Chair in Macroeconomics! · PDF fileContents 1 Problem 1 (The Real Business Cycle Model) Markus Roth (Advanced Macroeconomics) Problem set 6 February

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)


Problem 1 (The Real Business Cycle Model)

Contents

1 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



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.



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



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.



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)



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)



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)).



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.



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.


LHS ≃ EtC(ct − c).

• The right hand side is approximated by

RHS ≃ CβR+ CβR(ct + rt+1).



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.


RHS ≃ 1− δ+(1− α)

(

AN

K

)α

+ α(1− α)

(

AN

K

)α

(at+1+ nt+1− kt+1)



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.



Log-linearization 6


LHS ≃ Cθ(1−N)−γ + Cθ[(1−N)−γct − (1−N)−γ−1nt].


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)



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

.



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

.



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

.



Matrix C

• Finally matrix C is a column vector only and is given by

000001

.



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.



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.



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].


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.


Documents

Problem set 6 - Welcome to the Chair in Macroeconomics! · PDF fileContents 1 Problem 1 (The Real Business Cycle Model) Markus Roth (Advanced Macroeconomics) Problem set 6 February