Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann...

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Business Cycles Problem Set 4, Exercise 2 Business Cycles Problem Set 4, Exercise 2 F. Ferroni 1 1 Banque de France, February 2, 2011

Transcript of Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann...

Page 1: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Business CyclesProblem Set 4, Exercise 2

F. Ferroni1

1Banque de France,

February 2, 2011

Page 2: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Preliminary on DSGE models

Agents in the economy optimize. Compute FOC

we obtain a set of optimality equations

Etf (xt+1, xt, zt+1, zt) = 0

zt = g(zt−1, εt) εt ∼ N(0, σ)

where xt is a vector of endogenous variables and zt is a vector ofexogenous processes

We want to express the endogenous variables as a function ofexogenous ones, i.e. solve the model

xt = h(zt)

Analytical solution of h are not implementable. Log linearize theequilibrium conditions around a point, the steady state, and solvenumerically the linearized sistem

Page 3: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Preliminary on DSGE models

Agents in the economy optimize. Compute FOC

we obtain a set of optimality equations

Etf (xt+1, xt, zt+1, zt) = 0

zt = g(zt−1, εt) εt ∼ N(0, σ)

where xt is a vector of endogenous variables and zt is a vector ofexogenous processes

We want to express the endogenous variables as a function ofexogenous ones, i.e. solve the model

xt = h(zt)

Analytical solution of h are not implementable. Log linearize theequilibrium conditions around a point, the steady state, and solvenumerically the linearized sistem

Page 4: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Preliminary on DSGE models

Agents in the economy optimize. Compute FOC

we obtain a set of optimality equations

Etf (xt+1, xt, zt+1, zt) = 0

zt = g(zt−1, εt) εt ∼ N(0, σ)

where xt is a vector of endogenous variables and zt is a vector ofexogenous processes

We want to express the endogenous variables as a function ofexogenous ones, i.e. solve the model

xt = h(zt)

Analytical solution of h are not implementable. Log linearize theequilibrium conditions around a point, the steady state, and solvenumerically the linearized sistem

Page 5: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Preliminary on DSGE models

Agents in the economy optimize. Compute FOC

we obtain a set of optimality equations

Etf (xt+1, xt, zt+1, zt) = 0

zt = g(zt−1, εt) εt ∼ N(0, σ)

where xt is a vector of endogenous variables and zt is a vector ofexogenous processes

We want to express the endogenous variables as a function ofexogenous ones, i.e. solve the model

xt = h(zt)

Analytical solution of h are not implementable. Log linearize theequilibrium conditions around a point, the steady state, and solvenumerically the linearized sistem

Page 6: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

The RBC model

max E0

∞∑t=0

βt [ln ct + θ ln(1− ht)]

s.t. ezt kαt h1−αt + (1− δ)kt ≥ ct + kt+1

where 0 < β < 1, 0 < α < 1, 0 < δ < 1, θ is the Frish elasticity, and k0 isgiven. Assume further

zt = ρzt−1 + εt εt ∼ N(0, σ)

Page 7: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

The RBC model

max E0

∞∑t=0

βt [ln ct + θ ln(1− ht)]

s.t. ezt kαt h1−αt + (1− δ)kt ≥ ct + kt+1

where 0 < β < 1, 0 < α < 1, 0 < δ < 1, θ is the Frish elasticity, and k0 isgiven. Assume further

zt = ρzt−1 + εt εt ∼ N(0, σ)

Page 8: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

The RBC model

max E0

∞∑t=0

βt [ln ct + θ ln(1− ht)]

s.t. ezt kαt h1−αt + (1− δ)kt ≥ ct + kt+1

where 0 < β < 1, 0 < α < 1, 0 < δ < 1, θ is the Frish elasticity, and k0 isgiven. Assume further

zt = ρzt−1 + εt εt ∼ N(0, σ)

Page 9: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 1

Derive the first order condition for ct, ht, kt+1

Write the Lagrangian

E0

∞∑t=0

βt[ln ct + θ ln(1− ht) + λt(ezt kαt h1−α

t + (1− δ)kt − ct − kt+1)]=

E0([ln c0 + θ ln(1− h0) + λt(ez0 kα0 h1−α

0 + (1− δ)k0 − c0 − k1)]+ ...

βt[ln ct + θ ln(1− ht) + λt(ezt kαt h1−α

t + (1− δ)kt − ct − kt+1)]+

βt+1[ln ct+1 + θ ln(1− ht+1) + λt+1(ezt+1 kαt+1h1−α

t+1 + (1− δ)kt+1 − ct+1 − kt+2)]

+ ...)

Take derivatives wrt ct, ht, kt+1

Page 10: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 1

Derive the first order condition for ct, ht, kt+1

Write the Lagrangian

E0

∞∑t=0

βt[ln ct + θ ln(1− ht) + λt(ezt kαt h1−α

t + (1− δ)kt − ct − kt+1)]=

E0([ln c0 + θ ln(1− h0) + λt(ez0 kα0 h1−α

0 + (1− δ)k0 − c0 − k1)]+ ...

βt[ln ct + θ ln(1− ht) + λt(ezt kαt h1−α

t + (1− δ)kt − ct − kt+1)]+

βt+1[ln ct+1 + θ ln(1− ht+1) + λt+1(ezt+1 kαt+1h1−α

t+1 + (1− δ)kt+1 − ct+1 − kt+2)]

+ ...)

Take derivatives wrt ct, ht, kt+1

Page 11: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 1

Derive the first order condition for ct, ht, kt+1

Write the Lagrangian

E0

∞∑t=0

βt[ln ct + θ ln(1− ht) + λt(ezt kαt h1−α

t + (1− δ)kt − ct − kt+1)]=

E0([ln c0 + θ ln(1− h0) + λt(ez0 kα0 h1−α

0 + (1− δ)k0 − c0 − k1)]+ ...

βt[ln ct + θ ln(1− ht) + λt(ezt kαt h1−α

t + (1− δ)kt − ct − kt+1)]+

βt+1[ln ct+1 + θ ln(1− ht+1) + λt+1(ezt+1 kαt+1h1−α

t+1 + (1− δ)kt+1 − ct+1 − kt+2)]

+ ...)

Take derivatives wrt ct, ht, kt+1

Page 12: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 1

wrt ct

1/ct − λt = 0

wrt ht

− θ

1− ht+ (1− α)λtezt kαt h−αt = 0

wrt kt+1

−λt + βEtλt+1

(αezt+1 kα−1

t+1 h1−αt+1 + (1− δ)

)= 0

Page 13: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 1

wrt ct

1/ct − λt = 0

wrt ht

− θ

1− ht+ (1− α)λtezt kαt h−αt = 0

wrt kt+1

−λt + βEtλt+1

(αezt+1 kα−1

t+1 h1−αt+1 + (1− δ)

)= 0

Page 14: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 1

wrt ct

1/ct − λt = 0

wrt ht

− θ

1− ht+ (1− α)λtezt kαt h−αt = 0

wrt kt+1

−λt + βEtλt+1

(αezt+1 kα−1

t+1 h1−αt+1 + (1− δ)

)= 0

Page 15: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Substitute out the lagrangian multiplier and we get

θct

1− ht= (1− α)ezt kαt h−αt

1 = βEtct

ct+1

(αezt+1 kαt+1h−αt+1 + (1− δ)

)

Introduce the following variables

yt = ezt kαt h1−αt

it = kt+1 − (1− δ)kt

rt = αyt

kt

wt = (1− α) yt

ht

Notice also that

ezt kαt h−αt =yt

ht

ezt kα−1t h1−α

t =yt

kt

Page 16: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Substitute out the lagrangian multiplier and we get

θct

1− ht= (1− α)ezt kαt h−αt

1 = βEtct

ct+1

(αezt+1 kαt+1h−αt+1 + (1− δ)

)Introduce the following variables

yt = ezt kαt h1−αt

it = kt+1 − (1− δ)kt

rt = αyt

kt

wt = (1− α) yt

ht

Notice also that

ezt kαt h−αt =yt

ht

ezt kα−1t h1−α

t =yt

kt

Page 17: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Substitute out the lagrangian multiplier and we get

θct

1− ht= (1− α)ezt kαt h−αt

1 = βEtct

ct+1

(αezt+1 kαt+1h−αt+1 + (1− δ)

)Introduce the following variables

yt = ezt kαt h1−αt

it = kt+1 − (1− δ)kt

rt = αyt

kt

wt = (1− α) yt

ht

Notice also that

ezt kαt h−αt =yt

ht

ezt kα−1t h1−α

t =yt

kt

Page 18: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

Equilibrium Condition

wt =θct

1− ht(1)

1 = βEtct

ct+1(rt+1 + (1− δ)) (2)

yt = ezt kαt h1−αt (3)

it = kt+1 − (1− δ)kt (4)

rt = αyt

kt(5)

wt = (1− α) yt

ht(6)

yt = ct + it (7)

7 endogenous variables, ct, ht, kt+1, yt, rt,wt, 7 equations. 1 exogenous.

Compute the non stochastic steady state, i.e. εt = 0 for all t. Thus z = 0

Page 19: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

Equilibrium Condition

wt =θct

1− ht(1)

1 = βEtct

ct+1(rt+1 + (1− δ)) (2)

yt = ezt kαt h1−αt (3)

it = kt+1 − (1− δ)kt (4)

rt = αyt

kt(5)

wt = (1− α) yt

ht(6)

yt = ct + it (7)

7 endogenous variables, ct, ht, kt+1, yt, rt,wt, 7 equations. 1 exogenous.

Compute the non stochastic steady state, i.e. εt = 0 for all t. Thus z = 0

Page 20: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

Equilibrium Condition

wt =θct

1− ht(1)

1 = βEtct

ct+1(rt+1 + (1− δ)) (2)

yt = ezt kαt h1−αt (3)

it = kt+1 − (1− δ)kt (4)

rt = αyt

kt(5)

wt = (1− α) yt

ht(6)

yt = ct + it (7)

7 endogenous variables, ct, ht, kt+1, yt, rt,wt, 7 equations. 1 exogenous.

Compute the non stochastic steady state, i.e. εt = 0 for all t. Thus z = 0

Page 21: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

from (2)

1 = β (r + (1− δ))r = 1/β + δ − 1

from (5)

y/k =1/β + δ − 1

α

from (3)

y/h = (y/k)α

α−1 =

(1/β + δ − 1

α

) αα−1

from (6)

w = (1− α)y/h = (1− α)(

1/β + δ − 1α

) αα−1

from (4)

i/y = δk/y =δα

1/β + δ − 1

Page 22: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

from (2)

1 = β (r + (1− δ))r = 1/β + δ − 1

from (5)

y/k =1/β + δ − 1

α

from (3)

y/h = (y/k)α

α−1 =

(1/β + δ − 1

α

) αα−1

from (6)

w = (1− α)y/h = (1− α)(

1/β + δ − 1α

) αα−1

from (4)

i/y = δk/y =δα

1/β + δ − 1

Page 23: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

from (2)

1 = β (r + (1− δ))r = 1/β + δ − 1

from (5)

y/k =1/β + δ − 1

α

from (3)

y/h = (y/k)α

α−1 =

(1/β + δ − 1

α

) αα−1

from (6)

w = (1− α)y/h = (1− α)(

1/β + δ − 1α

) αα−1

from (4)

i/y = δk/y =δα

1/β + δ − 1

Page 24: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

from (2)

1 = β (r + (1− δ))r = 1/β + δ − 1

from (5)

y/k =1/β + δ − 1

α

from (3)

y/h = (y/k)α

α−1 =

(1/β + δ − 1

α

) αα−1

from (6)

w = (1− α)y/h = (1− α)(

1/β + δ − 1α

) αα−1

from (4)

i/y = δk/y =δα

1/β + δ − 1

Page 25: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

from (2)

1 = β (r + (1− δ))r = 1/β + δ − 1

from (5)

y/k =1/β + δ − 1

α

from (3)

y/h = (y/k)α

α−1 =

(1/β + δ − 1

α

) αα−1

from (6)

w = (1− α)y/h = (1− α)(

1/β + δ − 1α

) αα−1

from (4)

i/y = δk/y =δα

1/β + δ − 1

Page 26: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

from (7)

c/y = 1− i/y = 1− δα

1/β + δ − 1

from (1)

(1− α)y/h =θc

1− h1− αθ

1− hh

= c/y = 1− δα

1/β + δ − 1

h =1

1 + θ1−α c/y

Page 27: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 2

from (7)

c/y = 1− i/y = 1− δα

1/β + δ − 1

from (1)

(1− α)y/h =θc

1− h1− αθ

1− hh

= c/y = 1− δα

1/β + δ − 1

h =1

1 + θ1−α c/y

Page 28: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Consider the following function

F(s1,t, s2,t, s3,t) = 0

The log linear approximation is

F1 s1 s1,t + F2 s2 s2,t + F3 s3 s3,t = 0

where sj,t = ln sj,t − ln sj for j = 1, 2, 3 and Fj is the derivative of sj

variables evaluated at the steady state. See Uhlig(1998).

Notice that in a neighborhood of the steady state

sj,t = ln sj,t − ln sj 'sj,t − sj

sj

Then the latter represent the percentage deviation of a variable from thesteady state.

Page 29: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Consider the following function

F(s1,t, s2,t, s3,t) = 0

The log linear approximation is

F1 s1 s1,t + F2 s2 s2,t + F3 s3 s3,t = 0

where sj,t = ln sj,t − ln sj for j = 1, 2, 3 and Fj is the derivative of sj

variables evaluated at the steady state. See Uhlig(1998).

Notice that in a neighborhood of the steady state

sj,t = ln sj,t − ln sj 'sj,t − sj

sj

Then the latter represent the percentage deviation of a variable from thesteady state.

Page 30: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Consider the following function

F(s1,t, s2,t, s3,t) = 0

The log linear approximation is

F1 s1 s1,t + F2 s2 s2,t + F3 s3 s3,t = 0

where sj,t = ln sj,t − ln sj for j = 1, 2, 3 and Fj is the derivative of sj

variables evaluated at the steady state. See Uhlig(1998).

Notice that in a neighborhood of the steady state

sj,t = ln sj,t − ln sj 'sj,t − sj

sj

Then the latter represent the percentage deviation of a variable from thesteady state.

Page 31: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (1)

(1− α)yt/ht =θct

1− ht

(1− α)1/hyyt − (1− α) yh2 hht = θ

c1− h

ct + θc

(1− h)2 hht

since (1− α)y/h = θc1−h , we get

yt − ht = ct +h

1− hht

yt = ct +1

1− hht

Eq (2)

1 = βEtct

ct+1(rt+1 + (1− δ))

0 = Et (β(r + 1− δ)(ct − ct+1) + βrrt+1)

0 = Et (ct − ct+1 + βrrt+1)

since at the steady state β(r + 1− δ) = 1

Page 32: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (1)

(1− α)yt/ht =θct

1− ht

(1− α)1/hyyt − (1− α) yh2 hht = θ

c1− h

ct + θc

(1− h)2 hht

since (1− α)y/h = θc1−h , we get

yt − ht = ct +h

1− hht

yt = ct +1

1− hht

Eq (2)

1 = βEtct

ct+1(rt+1 + (1− δ))

0 = Et (β(r + 1− δ)(ct − ct+1) + βrrt+1)

0 = Et (ct − ct+1 + βrrt+1)

since at the steady state β(r + 1− δ) = 1

Page 33: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (1)

(1− α)yt/ht =θct

1− ht

(1− α)1/hyyt − (1− α) yh2 hht = θ

c1− h

ct + θc

(1− h)2 hht

since (1− α)y/h = θc1−h , we get

yt − ht = ct +h

1− hht

yt = ct +1

1− hht

Eq (2)

1 = βEtct

ct+1(rt+1 + (1− δ))

0 = Et (β(r + 1− δ)(ct − ct+1) + βrrt+1)

0 = Et (ct − ct+1 + βrrt+1)

since at the steady state β(r + 1− δ) = 1

Page 34: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (3)

yt = zt + αkt + (1− α)ht

Eq (5)

rt = yt − kt

Eq (6)

wt = yt − ht

Eq (4)

i it = kkt+1 − (1− δ)kkt

i/yit = k/ykt+1 − (1− δ)k/ykt

Eq (7)

yyt = cct + iit

yt = c/yct + i/yit

Page 35: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (3)

yt = zt + αkt + (1− α)ht

Eq (5)

rt = yt − kt

Eq (6)

wt = yt − ht

Eq (4)

i it = kkt+1 − (1− δ)kkt

i/yit = k/ykt+1 − (1− δ)k/ykt

Eq (7)

yyt = cct + iit

yt = c/yct + i/yit

Page 36: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (3)

yt = zt + αkt + (1− α)ht

Eq (5)

rt = yt − kt

Eq (6)

wt = yt − ht

Eq (4)

i it = kkt+1 − (1− δ)kkt

i/yit = k/ykt+1 − (1− δ)k/ykt

Eq (7)

yyt = cct + iit

yt = c/yct + i/yit

Page 37: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (3)

yt = zt + αkt + (1− α)ht

Eq (5)

rt = yt − kt

Eq (6)

wt = yt − ht

Eq (4)

i it = kkt+1 − (1− δ)kkt

i/yit = k/ykt+1 − (1− δ)k/ykt

Eq (7)

yyt = cct + iit

yt = c/yct + i/yit

Page 38: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Log linearization

Eq (3)

yt = zt + αkt + (1− α)ht

Eq (5)

rt = yt − kt

Eq (6)

wt = yt − ht

Eq (4)

i it = kkt+1 − (1− δ)kkt

i/yit = k/ykt+1 − (1− δ)k/ykt

Eq (7)

yyt = cct + iit

yt = c/yct + i/yit

Page 39: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Sum up

yt = ct +1

1− hht

0 = Et (ct − ct+1 + βrrt+1)

yt = zt + αkt + (1− α)ht

rt = yt − kt

wt = yt − ht

i/yit = k/ykt+1 − (1− δ)k/ykt

yt = c/yct + i/yit

and

zt = ρzt−1 + εt εt ∼ N(0, σ)

Page 40: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 3, Sum up

yt = ct +1

1− hht

0 = Et (ct − ct+1 + βrrt+1)

yt = zt + αkt + (1− α)ht

rt = yt − kt

wt = yt − ht

i/yit = k/ykt+1 − (1− δ)k/ykt

yt = c/yct + i/yit

and

zt = ρzt−1 + εt εt ∼ N(0, σ)

Page 41: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

Let xt = [ct, ht, kt+1, yt, it, rt,wt], show that the system satisfies

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt]

zt+1 = Nzt + εt+1

8 LINEAR equations, 7 endogenous variables and 1 exogenous. Beinglinear, the latter system can be written in matrix format.

Give an expression for F,G,H,L,M

Page 42: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

Let xt = [ct, ht, kt+1, yt, it, rt,wt], show that the system satisfies

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt]

zt+1 = Nzt + εt+1

8 LINEAR equations, 7 endogenous variables and 1 exogenous. Beinglinear, the latter system can be written in matrix format.

Give an expression for F,G,H,L,M

Page 43: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

Let xt = [ct, ht, kt+1, yt, it, rt,wt], show that the system satisfies

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt]

zt+1 = Nzt + εt+1

8 LINEAR equations, 7 endogenous variables and 1 exogenous. Beinglinear, the latter system can be written in matrix format.

Give an expression for F,G,H,L,M

Page 44: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

xt+1 = [ct+1, ht+1, kt+2, yt+1, it+1, rt+1,wt+1]

F =

0 0 0 0 0 0 0−1 0 0 0 0 βr 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

xt = [ct, ht, kt+1, yt, it, rt,wt]

G =

1 1/(1− h) 0 −1 0 0 01 0 0 0 0 0 00 1− α 0 −1 0 0 00 0 0 1 0 −1 00 −1 0 1 0 0 −10 0 k/y 0 i/y 0 0

c/y 0 0 −1 i/y 0 0

Page 45: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

xt+1 = [ct+1, ht+1, kt+2, yt+1, it+1, rt+1,wt+1]

F =

0 0 0 0 0 0 0−1 0 0 0 0 βr 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

xt = [ct, ht, kt+1, yt, it, rt,wt]

G =

1 1/(1− h) 0 −1 0 0 01 0 0 0 0 0 00 1− α 0 −1 0 0 00 0 0 1 0 −1 00 −1 0 1 0 0 −10 0 k/y 0 i/y 0 0

c/y 0 0 −1 i/y 0 0

Page 46: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

xt−1 = [ct−1, ht−1, kt, yt−1, it−1, rt−1,wt−1]

H =

0 0 0 0 0 0 00 0 0 0 0 0 00 0 α 0 0 0 00 0 −1 0 0 0 00 0 0 0 0 0 00 0 (1− δ)k/y 0 0 0 00 0 0 0 0 0 0

L =(

0 0 0 0 0 0 0)′

M =(

0 1 0 0 0 0 0)′

Page 47: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

xt−1 = [ct−1, ht−1, kt, yt−1, it−1, rt−1,wt−1]

H =

0 0 0 0 0 0 00 0 0 0 0 0 00 0 α 0 0 0 00 0 −1 0 0 0 00 0 0 0 0 0 00 0 (1− δ)k/y 0 0 0 00 0 0 0 0 0 0

L =

(0 0 0 0 0 0 0

)′

M =(

0 1 0 0 0 0 0)′

Page 48: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 4

xt−1 = [ct−1, ht−1, kt, yt−1, it−1, rt−1,wt−1]

H =

0 0 0 0 0 0 00 0 0 0 0 0 00 0 α 0 0 0 00 0 −1 0 0 0 00 0 0 0 0 0 00 0 (1− δ)k/y 0 0 0 00 0 0 0 0 0 0

L =

(0 0 0 0 0 0 0

)′M =

(0 1 0 0 0 0 0

)′

Page 49: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 5, Guess the solution

Guess that the solution takes the following form

xt = Pxt−1 + Qzt

and that P solves a quadratic equation

Plug the latter in our system and we get

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt] =

= Et

[F(P2xt−1 + PQzt + Qzt+1) + G(Pxt−1 + Qzt) + Hxt−1 + Lzt+1 + Mzt

]=

= Et[F(P2xt−1 + PQzt + QNzt + Qεt+1) + G(Pxt−1 + Qzt)

+ Hxt−1 + LNzt + Lεt+1 + Mzt] =

= xt−1(FP2 + GP + H) + zt(FPQ + FQN + GQ + LN + M)

the solution must hold for every xt−1 and zt.

first solve P2 + GP + H = 0 and get P

given P solve for Q using (FPQ + FQN + GQ + LN + M) = 0

Page 50: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 5, Guess the solution

Guess that the solution takes the following form

xt = Pxt−1 + Qzt

and that P solves a quadratic equation

Plug the latter in our system and we get

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt] =

= Et

[F(P2xt−1 + PQzt + Qzt+1) + G(Pxt−1 + Qzt) + Hxt−1 + Lzt+1 + Mzt

]=

= Et[F(P2xt−1 + PQzt + QNzt + Qεt+1) + G(Pxt−1 + Qzt)

+ Hxt−1 + LNzt + Lεt+1 + Mzt] =

= xt−1(FP2 + GP + H) + zt(FPQ + FQN + GQ + LN + M)

the solution must hold for every xt−1 and zt.

first solve P2 + GP + H = 0 and get P

given P solve for Q using (FPQ + FQN + GQ + LN + M) = 0

Page 51: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 5, Guess the solution

Guess that the solution takes the following form

xt = Pxt−1 + Qzt

and that P solves a quadratic equation

Plug the latter in our system and we get

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt] =

= Et

[F(P2xt−1 + PQzt + Qzt+1) + G(Pxt−1 + Qzt) + Hxt−1 + Lzt+1 + Mzt

]=

= Et[F(P2xt−1 + PQzt + QNzt + Qεt+1) + G(Pxt−1 + Qzt)

+ Hxt−1 + LNzt + Lεt+1 + Mzt] =

= xt−1(FP2 + GP + H) + zt(FPQ + FQN + GQ + LN + M)

the solution must hold for every xt−1 and zt.

first solve P2 + GP + H = 0 and get P

given P solve for Q using (FPQ + FQN + GQ + LN + M) = 0

Page 52: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 5, Guess the solution

Guess that the solution takes the following form

xt = Pxt−1 + Qzt

and that P solves a quadratic equation

Plug the latter in our system and we get

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt] =

= Et

[F(P2xt−1 + PQzt + Qzt+1) + G(Pxt−1 + Qzt) + Hxt−1 + Lzt+1 + Mzt

]=

= Et[F(P2xt−1 + PQzt + QNzt + Qεt+1) + G(Pxt−1 + Qzt)

+ Hxt−1 + LNzt + Lεt+1 + Mzt] =

= xt−1(FP2 + GP + H) + zt(FPQ + FQN + GQ + LN + M)

the solution must hold for every xt−1 and zt.

first solve P2 + GP + H = 0 and get P

given P solve for Q using (FPQ + FQN + GQ + LN + M) = 0

Page 53: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 5, Guess the solution

Guess that the solution takes the following form

xt = Pxt−1 + Qzt

and that P solves a quadratic equation

Plug the latter in our system and we get

0 = Et [Fxt+1 + Gxt + Hxt−1 + Lzt+1 + Mzt] =

= Et

[F(P2xt−1 + PQzt + Qzt+1) + G(Pxt−1 + Qzt) + Hxt−1 + Lzt+1 + Mzt

]=

= Et[F(P2xt−1 + PQzt + QNzt + Qεt+1) + G(Pxt−1 + Qzt)

+ Hxt−1 + LNzt + Lεt+1 + Mzt] =

= xt−1(FP2 + GP + H) + zt(FPQ + FQN + GQ + LN + M)

the solution must hold for every xt−1 and zt.

first solve P2 + GP + H = 0 and get P

given P solve for Q using (FPQ + FQN + GQ + LN + M) = 0

Page 54: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc

Page 55: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc

Page 56: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc

Page 57: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc

Page 58: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc

Page 59: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc

Page 60: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc

Page 61: Business Cycles Problem Set 4, Exercise 2econ.sciences-po.fr/sites/default/files/file/yann algan/M1EPP_Chap4... · Business Cycles Problem Set 4, Exercise 2 Preliminary on DSGE models

Business Cycles Problem Set 4, Exercise 2

Question 6, Dynare

Declare the variablesvar y c k h i r w z;

Declare the exogenousvarexo ez;

declare the parametersparameters cy ky alpha beta delta rho hbar theta iy

r0;

Assign numerical values to the parameters

Write down the equilibrium conditionsmodel; ... end;

set the standard deviation of the shocksshocks; var ez; stderr 0.0072; end;

simulate data and compute impulse responsestoch simul(order=0,periods=115,irf=15);

save all in a rbc.mod file, type in matlab dynare rbc