Post on 03-Feb-2022
Notes on the Real Business Cycle Model
Pengfei Wang
Hong Kong University of Science and Technology
2010
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 1 / 46
Introduction: Basic Facts about Business Cycle
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 2 / 46
Introduction: Basic Facts about Business Cycle 2
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 3 / 46
Introduction: Basic Facts about Business Cycle 2
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 4 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theoryClean-up theorySunspots theory
Need to distinguish
Source of ShocksPropagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theory
Clean-up theorySunspots theory
Need to distinguish
Source of ShocksPropagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theoryClean-up theory
Sunspots theory
Need to distinguish
Source of ShocksPropagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theoryClean-up theorySunspots theory
Need to distinguish
Source of ShocksPropagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theoryClean-up theorySunspots theory
Need to distinguish
Source of ShocksPropagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theoryClean-up theorySunspots theory
Need to distinguish
Source of Shocks
Propagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theoryClean-up theorySunspots theory
Need to distinguish
Source of ShocksPropagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Introduction: The central questions
What cuases business cycles?
Multiplier-accelerator theoryClean-up theorySunspots theory
Need to distinguish
Source of ShocksPropagation Mechanisms
How Should Government Policy Respond to Business Cycles?
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 5 / 46
Two Major Schools of Thoughts:
The Classical School�>Doctrine: Supply determines demand
supply shocks are the major source of business cycles.
Supply shocks include shocks to technology, to endowment, to costsof production, investment e¢ ciency,etc.
Propagation mechanisms include adjustment costs, capitalaccumulation, time to build,etc.
Policy recommendation: Do nothing.
Features: Very rigorous but against common sense.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 6 / 46
Two Major Schools of Thoughts:
The Classical School�>Doctrine: Supply determines demand
supply shocks are the major source of business cycles.
Supply shocks include shocks to technology, to endowment, to costsof production, investment e¢ ciency,etc.
Propagation mechanisms include adjustment costs, capitalaccumulation, time to build,etc.
Policy recommendation: Do nothing.
Features: Very rigorous but against common sense.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 6 / 46
Two Major Schools of Thoughts:
The Classical School�>Doctrine: Supply determines demand
supply shocks are the major source of business cycles.
Supply shocks include shocks to technology, to endowment, to costsof production, investment e¢ ciency,etc.
Propagation mechanisms include adjustment costs, capitalaccumulation, time to build,etc.
Policy recommendation: Do nothing.
Features: Very rigorous but against common sense.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 6 / 46
Two Major Schools of Thoughts:
The Classical School�>Doctrine: Supply determines demand
supply shocks are the major source of business cycles.
Supply shocks include shocks to technology, to endowment, to costsof production, investment e¢ ciency,etc.
Propagation mechanisms include adjustment costs, capitalaccumulation, time to build,etc.
Policy recommendation: Do nothing.
Features: Very rigorous but against common sense.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 6 / 46
Two Major Schools of Thoughts:
The Classical School�>Doctrine: Supply determines demand
supply shocks are the major source of business cycles.
Supply shocks include shocks to technology, to endowment, to costsof production, investment e¢ ciency,etc.
Propagation mechanisms include adjustment costs, capitalaccumulation, time to build,etc.
Policy recommendation: Do nothing.
Features: Very rigorous but against common sense.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 6 / 46
Two Major Schools of Thoughts:
The Keynesian School�> Doctrine: Demand determines supply
demand shocks are the major source of business cycles.
Demand shocks include shocks to consumption, to investment, togovernment spending, to exports, and to money supply;
Shocks to consumption or investment also include extrinsicuncertainty (expectations, animal spirits, sunspots).
Propagation mechanisms include interest rates, portfolio allocations,asset bubbles (including the stock market), banking and �nancialcrises, and international trade and currency crises.
Policy recommendation: Intervene
Features: Appealing to common sense but very vague and imprecise.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 7 / 46
Two Major Schools of Thoughts:
The Keynesian School�> Doctrine: Demand determines supply
demand shocks are the major source of business cycles.
Demand shocks include shocks to consumption, to investment, togovernment spending, to exports, and to money supply;
Shocks to consumption or investment also include extrinsicuncertainty (expectations, animal spirits, sunspots).
Propagation mechanisms include interest rates, portfolio allocations,asset bubbles (including the stock market), banking and �nancialcrises, and international trade and currency crises.
Policy recommendation: Intervene
Features: Appealing to common sense but very vague and imprecise.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 7 / 46
Two Major Schools of Thoughts:
The Keynesian School�> Doctrine: Demand determines supply
demand shocks are the major source of business cycles.
Demand shocks include shocks to consumption, to investment, togovernment spending, to exports, and to money supply;
Shocks to consumption or investment also include extrinsicuncertainty (expectations, animal spirits, sunspots).
Propagation mechanisms include interest rates, portfolio allocations,asset bubbles (including the stock market), banking and �nancialcrises, and international trade and currency crises.
Policy recommendation: Intervene
Features: Appealing to common sense but very vague and imprecise.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 7 / 46
Two Major Schools of Thoughts:
The Keynesian School�> Doctrine: Demand determines supply
demand shocks are the major source of business cycles.
Demand shocks include shocks to consumption, to investment, togovernment spending, to exports, and to money supply;
Shocks to consumption or investment also include extrinsicuncertainty (expectations, animal spirits, sunspots).
Propagation mechanisms include interest rates, portfolio allocations,asset bubbles (including the stock market), banking and �nancialcrises, and international trade and currency crises.
Policy recommendation: Intervene
Features: Appealing to common sense but very vague and imprecise.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 7 / 46
Two Major Schools of Thoughts:
The Keynesian School�> Doctrine: Demand determines supply
demand shocks are the major source of business cycles.
Demand shocks include shocks to consumption, to investment, togovernment spending, to exports, and to money supply;
Shocks to consumption or investment also include extrinsicuncertainty (expectations, animal spirits, sunspots).
Propagation mechanisms include interest rates, portfolio allocations,asset bubbles (including the stock market), banking and �nancialcrises, and international trade and currency crises.
Policy recommendation: Intervene
Features: Appealing to common sense but very vague and imprecise.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 7 / 46
Two Major Schools of Thoughts:
The Keynesian School�> Doctrine: Demand determines supply
demand shocks are the major source of business cycles.
Demand shocks include shocks to consumption, to investment, togovernment spending, to exports, and to money supply;
Shocks to consumption or investment also include extrinsicuncertainty (expectations, animal spirits, sunspots).
Propagation mechanisms include interest rates, portfolio allocations,asset bubbles (including the stock market), banking and �nancialcrises, and international trade and currency crises.
Policy recommendation: Intervene
Features: Appealing to common sense but very vague and imprecise.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 7 / 46
Key Assumptions:
prices adjust instantaneously to clear markets
rational expectations
perfect competition
perfect risk sharing
no asymmetric information
no externalities
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 8 / 46
A Benchmark Model:
A social planner (or representative agent) chooses paths of consumption,investment and employment (hours) to solve
objective function
maxfct ,kt+1,ntg∞
t=0
E0∞
∑t=0
βt [log ct + γ log (1� nt )] (1)
Resource Constraint
ct + kt+1 � (1� δ) kt = Atkαt n1�αt , (2)
Shock ProcesslogAt = ρ logAt�1 + εt . (3)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 9 / 46
A Benchmark Model:
A social planner (or representative agent) chooses paths of consumption,investment and employment (hours) to solve
objective function
maxfct ,kt+1,ntg∞
t=0
E0∞
∑t=0
βt [log ct + γ log (1� nt )] (1)
Resource Constraint
ct + kt+1 � (1� δ) kt = Atkαt n1�αt , (2)
Shock ProcesslogAt = ρ logAt�1 + εt . (3)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 9 / 46
A Benchmark Model:
A social planner (or representative agent) chooses paths of consumption,investment and employment (hours) to solve
objective function
maxfct ,kt+1,ntg∞
t=0
E0∞
∑t=0
βt [log ct + γ log (1� nt )] (1)
Resource Constraint
ct + kt+1 � (1� δ) kt = Atkαt n1�αt , (2)
Shock ProcesslogAt = ρ logAt�1 + εt . (3)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 9 / 46
The Lagrangian:
The Lagrangian is given by
L = E0
�∑∞t=0 βt [log ct + γ log (1� nt )]
+λt�Atkα
t n1�αt � ct � kt+1 + (1� δ) kt
� � . (4)
state variables in period tkt ;At (5)
choice variables in period t
ct , nt ,λt , kt+1 (6)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 10 / 46
The Lagrangian:
The Lagrangian is given by
L = E0
�∑∞t=0 βt [log ct + γ log (1� nt )]
+λt�Atkα
t n1�αt � ct � kt+1 + (1� δ) kt
� � . (4)
state variables in period tkt ;At (5)
choice variables in period t
ct , nt ,λt , kt+1 (6)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 10 / 46
The Lagrangian:
The Lagrangian is given by
L = E0
�∑∞t=0 βt [log ct + γ log (1� nt )]
+λt�Atkα
t n1�αt � ct � kt+1 + (1� δ) kt
� � . (4)
state variables in period tkt ;At (5)
choice variables in period t
ct , nt ,λt , kt+1 (6)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 10 / 46
The First Order Conditions:
with respect to consumption :
1ct= λt (7)
with respect to labor:
γ
1� nt= λt
�(1� α)Atkα
t n�αt
�(8)
with respect to kt+1
λt = βEt�λt+1
�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
��(9)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 11 / 46
The First Order Conditions:
with respect to consumption :
1ct= λt (7)
with respect to labor:
γ
1� nt= λt
�(1� α)Atkα
t n�αt
�(8)
with respect to kt+1
λt = βEt�λt+1
�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
��(9)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 11 / 46
The First Order Conditions:
with respect to consumption :
1ct= λt (7)
with respect to labor:
γ
1� nt= λt
�(1� α)Atkα
t n�αt
�(8)
with respect to kt+1
λt = βEt�λt+1
�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
��(9)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 11 / 46
The First Order Conditions (continued):
with respect to λt :
ct + kt+1 � (1� δ) kt = Atkαt n1�αt , (10)
plus a standard transversality condition:
limT!∞
E0βT λT kT+1 = 0, (11)
and the law of motion for technology
logAt = ρ logAt�1 + εt . (12)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 12 / 46
The First Order Conditions (continued):
with respect to λt :
ct + kt+1 � (1� δ) kt = Atkαt n1�αt , (10)
plus a standard transversality condition:
limT!∞
E0βT λT kT+1 = 0, (11)
and the law of motion for technology
logAt = ρ logAt�1 + εt . (12)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 12 / 46
The First Order Conditions (continued):
with respect to λt :
ct + kt+1 � (1� δ) kt = Atkαt n1�αt , (10)
plus a standard transversality condition:
limT!∞
E0βT λT kT+1 = 0, (11)
and the law of motion for technology
logAt = ρ logAt�1 + εt . (12)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 12 / 46
Equilibrium:
An equilibrium is a set of decision rules:
xt = x (kt ,At )
for x = fct , kt+1, nt ,λtg such that equations (7)-(12) are satis�ed.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 13 / 46
Steady State :
In a steady state, (9) implies
1 = βhαyk+ 1� δ
i(13)
orky=
αβ
1� β (1� δ). (14)
and (10) impliescy= 1� δ
ky. (15)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 14 / 46
Steady State :
In a steady state, (9) implies
1 = βhαyk+ 1� δ
i(13)
orky=
αβ
1� β (1� δ). (14)
and (10) impliescy= 1� δ
ky. (15)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 14 / 46
Steady State :
In a steady state, (9) implies
1 = βhαyk+ 1� δ
i(13)
orky=
αβ
1� β (1� δ). (14)
and (10) impliescy= 1� δ
ky. (15)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 14 / 46
Steady State (continued) :
Hence, the great ratios are given by
iy
=δαβ
1� β (1� δ)(16)
cy
= 1� δαβ
1� β (1� δ)(17)
ky
=αβ
1� β (1� δ). (18)
Note that the steady-state rate of saving is given by
s� =iy=
δαβ
1� β (1� δ). (19)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 15 / 46
Steady State (continued) :
Hence, the great ratios are given by
iy
=δαβ
1� β (1� δ)(16)
cy
= 1� δαβ
1� β (1� δ)(17)
ky
=αβ
1� β (1� δ). (18)
Note that the steady-state rate of saving is given by
s� =iy=
δαβ
1� β (1� δ). (19)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 15 / 46
Steady State (continued) :
To solve for the steady-state levels, we note that (7) and (8) imply
γn1� n = (1� α)
yc
(20)
or
n� =(1� α) yc
γ+ (1� α) yc< 1. (21)
and by yk = A
� nk
�1�α, we have
k� =
�A�ky
�� 11�α
n� (22)
y � = A (k�)α (n�)1�α ; i� = s�y �; c� = (1� s�) y � (23)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 16 / 46
Steady State (continued) :
To solve for the steady-state levels, we note that (7) and (8) imply
γn1� n = (1� α)
yc
(20)
or
n� =(1� α) yc
γ+ (1� α) yc< 1. (21)
and by yk = A
� nk
�1�α, we have
k� =
�A�ky
�� 11�α
n� (22)
y � = A (k�)α (n�)1�α ; i� = s�y �; c� = (1� s�) y � (23)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 16 / 46
Steady State (continued) :
To solve for the steady-state levels, we note that (7) and (8) imply
γn1� n = (1� α)
yc
(20)
or
n� =(1� α) yc
γ+ (1� α) yc< 1. (21)
and by yk = A
� nk
�1�α, we have
k� =
�A�ky
�� 11�α
n� (22)
y � = A (k�)α (n�)1�α ; i� = s�y �; c� = (1� s�) y � (23)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 16 / 46
A Decentralized Version-The �rms :
Aggregate Production Technology:
Yt = AtK αt (ntLt )
1�α , (24)
where n is hours per worker and L is the labor force (without loss ofgenerality, assuming its growth rate be zero).
The per-worker production function is given by
yt = Atkαt n1�αt . (25)
Pro�t maximization:
max [f (kt , nt )� wtnt � (rt + δ)kt ] (26)
which implies the following factor demand functions:
rt + δ = αAtkα�1t n1�α
t (27)
wt = (1� α)Atkαt n�αt . (28)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 17 / 46
A Decentralized Version-The �rms :
Aggregate Production Technology:
Yt = AtK αt (ntLt )
1�α , (24)
where n is hours per worker and L is the labor force (without loss ofgenerality, assuming its growth rate be zero).The per-worker production function is given by
yt = Atkαt n1�αt . (25)
Pro�t maximization:
max [f (kt , nt )� wtnt � (rt + δ)kt ] (26)
which implies the following factor demand functions:
rt + δ = αAtkα�1t n1�α
t (27)
wt = (1� α)Atkαt n�αt . (28)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 17 / 46
A Decentralized Version-The �rms :
Aggregate Production Technology:
Yt = AtK αt (ntLt )
1�α , (24)
where n is hours per worker and L is the labor force (without loss ofgenerality, assuming its growth rate be zero).The per-worker production function is given by
yt = Atkαt n1�αt . (25)
Pro�t maximization:
max [f (kt , nt )� wtnt � (rt + δ)kt ] (26)
which implies the following factor demand functions:
rt + δ = αAtkα�1t n1�α
t (27)
wt = (1� α)Atkαt n�αt . (28)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 17 / 46
A Decentralized Version-The �rms :
Aggregate Production Technology:
Yt = AtK αt (ntLt )
1�α , (24)
where n is hours per worker and L is the labor force (without loss ofgenerality, assuming its growth rate be zero).The per-worker production function is given by
yt = Atkαt n1�αt . (25)
Pro�t maximization:
max [f (kt , nt )� wtnt � (rt + δ)kt ] (26)
which implies the following factor demand functions:
rt + δ = αAtkα�1t n1�α
t (27)
wt = (1� α)Atkαt n�αt . (28)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 17 / 46
A Decentralized Version-The households :
A representative worker�s problem:
maxfct ,st+1,nst g∞
t=0
E0∞
∑t=0
βt [log ct + γ log (1� nst )] (29)
subject toct + st+1 = (1+ rt ) st + wtnst . (30)
First order conditions:
1ct
= λt (31)
γ
1� nst= λtwt
λt = βEtλt+1 (1+ rt+1) .
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 18 / 46
A Decentralized Version-The households :
A representative worker�s problem:
maxfct ,st+1,nst g∞
t=0
E0∞
∑t=0
βt [log ct + γ log (1� nst )] (29)
subject toct + st+1 = (1+ rt ) st + wtnst . (30)
First order conditions:
1ct
= λt (31)
γ
1� nst= λtwt
λt = βEtλt+1 (1+ rt+1) .
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 18 / 46
A Decentralized Version-The households :
A representative worker�s problem:
maxfct ,st+1,nst g∞
t=0
E0∞
∑t=0
βt [log ct + γ log (1� nst )] (29)
subject toct + st+1 = (1+ rt ) st + wtnst . (30)
First order conditions:
1ct
= λt (31)
γ
1� nst= λtwt
λt = βEtλt+1 (1+ rt+1) .
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 18 / 46
A Decentralized Version-Equilibrium :
Equilibrium: In equilibrium, prices clear the markets and supply meetsdemand:
st+1 = kt+1 (32)
nst = nt . (33)
Hence we have:
1ct
= λt (34)
γ
1� nt= λt (1� α)Atkα
t n�αt (35)
λt = βEtλt+1�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
�(36)
ct + kt+1 = (1� δ) kt + Atkαt n1�αt , (37)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 19 / 46
A Decentralized Version-Equilibrium :
Equilibrium: In equilibrium, prices clear the markets and supply meetsdemand:
st+1 = kt+1 (32)
nst = nt . (33)
Hence we have:
1ct
= λt (34)
γ
1� nt= λt (1� α)Atkα
t n�αt (35)
λt = βEtλt+1�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
�(36)
ct + kt+1 = (1� δ) kt + Atkαt n1�αt , (37)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 19 / 46
A special case :
Equations (3) and (4) become
1ct= βEt
�1ct+1
αyt+1kt+1
�(38)
ct + kt+1 = Atkαt n1�αt . (39)
Guess the decision rule:ct = ξyt . (40)
Then we have,
1yt= βEt
�1yt+1
αyt+1kt+1
�= βαEt
1kt+1
, (41)
which implieskt+1 = βαyt . (42)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 20 / 46
A special case :
Equations (3) and (4) become
1ct= βEt
�1ct+1
αyt+1kt+1
�(38)
ct + kt+1 = Atkαt n1�αt . (39)
Guess the decision rule:ct = ξyt . (40)
Then we have,
1yt= βEt
�1yt+1
αyt+1kt+1
�= βαEt
1kt+1
, (41)
which implieskt+1 = βαyt . (42)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 20 / 46
A special case :
Equations (3) and (4) become
1ct= βEt
�1ct+1
αyt+1kt+1
�(38)
ct + kt+1 = Atkαt n1�αt . (39)
Guess the decision rule:ct = ξyt . (40)
Then we have,
1yt= βEt
�1yt+1
αyt+1kt+1
�= βαEt
1kt+1
, (41)
which implieskt+1 = βαyt . (42)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 20 / 46
A special case continued:
Substituting this into (39) gives
ct = (1� βα) yt . (43)
(8) then implies
γ
1� nt=
1(1� βα) yt
�(1� α)
ytnt
�=
1� α
1� βα
1nt, (44)
which implies
nt =1�α1�βα
γ+ 1�α1�βα
2 (0, 1). (45)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 21 / 46
A special case continued:
Substituting this into (39) gives
ct = (1� βα) yt . (43)
(8) then implies
γ
1� nt=
1(1� βα) yt
�(1� α)
ytnt
�=
1� α
1� βα
1nt, (44)
which implies
nt =1�α1�βα
γ+ 1�α1�βα
2 (0, 1). (45)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 21 / 46
A special case continued:
Substituting this into (39) gives
ct = (1� βα) yt . (43)
(8) then implies
γ
1� nt=
1(1� βα) yt
�(1� α)
ytnt
�=
1� α
1� βα
1nt, (44)
which implies
nt =1�α1�βα
γ+ 1�α1�βα
2 (0, 1). (45)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 21 / 46
A special case continued:
we have already obtained
nt =1�α1�βα
γ+ 1�α1�βα
2 (0, 1). (46)
Hence the decision rules are given by
yt = Atkαt n1�α (47)
ct = (1� βα)Atkαt n1�α (48)
kt+1 = βαAtkαt n1�α. (49)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 22 / 46
A special case continued:
we have already obtained
nt =1�α1�βα
γ+ 1�α1�βα
2 (0, 1). (46)
Hence the decision rules are given by
yt = Atkαt n1�α (47)
ct = (1� βα)Atkαt n1�α (48)
kt+1 = βαAtkαt n1�α. (49)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 22 / 46
A special case -Log linearization:
take log we have
yt = At + αkt (50)
ct = At + αkt (51)
kt+1 = At + αkt . (52)
Since
At = ρAt�1 + εt
=1
1� ρLεt =
∞
∑j=0
ρj εt�j , (53)
where L is a lag operator LX (t) = X (t � 1), LjX (t) = X (t � j)the decision rule for capital can be rewritten as a moving-averageprocess:
kt+1 =1
1� αLAt =
1(1� αL) (1� ρL)
εt ,
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 23 / 46
A special case -Log linearization:
take log we have
yt = At + αkt (50)
ct = At + αkt (51)
kt+1 = At + αkt . (52)
Since
At = ρAt�1 + εt
=1
1� ρLεt =
∞
∑j=0
ρj εt�j , (53)
where L is a lag operator LX (t) = X (t � 1), LjX (t) = X (t � j)
the decision rule for capital can be rewritten as a moving-averageprocess:
kt+1 =1
1� αLAt =
1(1� αL) (1� ρL)
εt ,
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 23 / 46
A special case -Log linearization:
take log we have
yt = At + αkt (50)
ct = At + αkt (51)
kt+1 = At + αkt . (52)
Since
At = ρAt�1 + εt
=1
1� ρLεt =
∞
∑j=0
ρj εt�j , (53)
where L is a lag operator LX (t) = X (t � 1), LjX (t) = X (t � j)the decision rule for capital can be rewritten as a moving-averageprocess:
kt+1 =1
1� αLAt =
1(1� αL) (1� ρL)
εt ,
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 23 / 46
A special case -Log linearization(continued):
or as an AR(2) process:
(1� αL) (1� ρL) kt+1 = εt (54)�1� (α+ ρ) L+ αρL2
�kt+1 = εt (55)
orkt+1 = (α+ ρ) kt � αρkt�1 + εt . (56)
Utilizing (50), we can also express consumption and output as ARMA(p, q) processes (p = 2, q = 0):
xt =1
1� ρLεt +
α
(1� αL) (1� ρL)εt�1 (57)
or
(1� αL) (1� ρL) xt = (1� αL) εt + αεt�1
= εt . (58)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 24 / 46
A special case -Log linearization(continued):
or as an AR(2) process:
(1� αL) (1� ρL) kt+1 = εt (54)�1� (α+ ρ) L+ αρL2
�kt+1 = εt (55)
orkt+1 = (α+ ρ) kt � αρkt�1 + εt . (56)
Utilizing (50), we can also express consumption and output as ARMA(p, q) processes (p = 2, q = 0):
xt =1
1� ρLεt +
α
(1� αL) (1� ρL)εt�1 (57)
or
(1� αL) (1� ρL) xt = (1� αL) εt + αεt�1
= εt . (58)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 24 / 46
A special case -Log linearization(continued):
or as an AR(2) process:
(1� αL) (1� ρL) kt+1 = εt (54)�1� (α+ ρ) L+ αρL2
�kt+1 = εt (55)
orkt+1 = (α+ ρ) kt � αρkt�1 + εt . (56)
Utilizing (50), we can also express consumption and output as ARMA(p, q) processes (p = 2, q = 0):
xt =1
1� ρLεt +
α
(1� αL) (1� ρL)εt�1 (57)
or
(1� αL) (1� ρL) xt = (1� αL) εt + αεt�1
= εt . (58)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 24 / 46
A special case -Log linearization(continued):
or as an AR(2) process:
(1� αL) (1� ρL) kt+1 = εt (54)�1� (α+ ρ) L+ αρL2
�kt+1 = εt (55)
orkt+1 = (α+ ρ) kt � αρkt�1 + εt . (56)
Utilizing (50), we can also express consumption and output as ARMA(p, q) processes (p = 2, q = 0):
xt =1
1� ρLεt +
α
(1� αL) (1� ρL)εt�1 (57)
or
(1� αL) (1� ρL) xt = (1� αL) εt + αεt�1
= εt . (58)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 24 / 46
A special case -impulse responses:
Consider the impulse responses of capital to a one unit increase intechnology at time t. Since
kt+1 = (α+ ρ) kt � αρkt�1 + εt , (59)
or using the state-space representation, we have�kt+1kt
�=
�α+ ρ �αρ1 0
��ktkt�1
�+
�1 00 0
��εtεt�1
�, (60)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 25 / 46
A special case -impulse responses:
Consider the impulse responses of capital to a one unit increase intechnology at time t. Since
kt+1 = (α+ ρ) kt � αρkt�1 + εt , (59)
or using the state-space representation, we have�kt+1kt
�=
�α+ ρ �αρ1 0
��ktkt�1
�+
�1 00 0
��εtεt�1
�, (60)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 25 / 46
Log linearization:
First-order conditions:1ct= λt (61)
γ
1� nt= λt
�(1� α)Atkα
t n�αt
�(62)
λt = βEt�λt+1
�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
��(63)
ct + kt+1 � (1� δ) kt = Atkαt n1�αt . (64)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 26 / 46
Log linearization (continued ):
Denote xt = xt�xx ' log(xt )� log(x) as the percentage change from
steady-state.
log-linearizing equationyt = xα
t (65)
yieldsyt = αxt (66)
log-linearizing equationyt = xα
1txβ2t (67)
yiedsyt = αx1t + βx2t (68)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 27 / 46
Log linearization (continued ):
Denote xt = xt�xx ' log(xt )� log(x) as the percentage change from
steady-state.
log-linearizing equationyt = xα
t (65)
yieldsyt = αxt (66)
log-linearizing equationyt = xα
1txβ2t (67)
yiedsyt = αx1t + βx2t (68)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 27 / 46
Log linearization (continued ):
Denote xt = xt�xx ' log(xt )� log(x) as the percentage change from
steady-state.
log-linearizing equationyt = xα
t (65)
yieldsyt = αxt (66)
log-linearizing equationyt = xα
1txβ2t (67)
yiedsyt = αx1t + βx2t (68)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 27 / 46
Log linearization (continued ):
log-linearizing equationyt = x1t + x2t (69)
yieldsyt =
x1x1 + x2
x1t +x2
x1 + x2x2t (70)
log-linearizing equation
yt =J
∑j=1xjt (71)
yieds
yt =J
∑j=1
�xjy
�xjt (72)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 28 / 46
Log linearization (continued ):
log-linearizing equationyt = x1t + x2t (69)
yieldsyt =
x1x1 + x2
x1t +x2
x1 + x2x2t (70)
log-linearizing equation
yt =J
∑j=1xjt (71)
yieds
yt =J
∑j=1
�xjy
�xjt (72)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 28 / 46
Log linearization (continued ):
log-linearizing equation
yt = Et [x1t+1 + x2t+1] (73)
yieldsyt =
x1x1 + x2
Et x1t+1 +x2
x1 + x2Et x2t+1 (74)
log-linearizing equationyt = Etxα
t+1 (75)
yiedsyt = αEt xt+1 (76)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 29 / 46
Log linearization (continued ):
log-linearizing equation
yt = Et [x1t+1 + x2t+1] (73)
yieldsyt =
x1x1 + x2
Et x1t+1 +x2
x1 + x2Et x2t+1 (74)
log-linearizing equationyt = Etxα
t+1 (75)
yiedsyt = αEt xt+1 (76)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 29 / 46
Log linearizing the f.o.cs:
Log linearization the equation
1ct= λt (77)
gives�ct = λt (78)
Log linearization the equation
γ
1� nt= λt (1� α)Atkα
t n�αt (79)
yiedsn
1� n nt = λt + At + αkt � αnt (80)
where sc = cy , si =
iy .
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 30 / 46
Log linearizing the f.o.cs:
Log linearization the equation
1ct= λt (77)
gives�ct = λt (78)
Log linearization the equation
γ
1� nt= λt (1� α)Atkα
t n�αt (79)
yiedsn
1� n nt = λt + At + αkt � αnt (80)
where sc = cy , si =
iy .
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 30 / 46
Log linearizing the f.o.cs(continued):
Log linearization the equation
λt = βEt�λt+1
�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
��(81)
yields
λt = Et�
λt+1 + (1� β (1� δ))�(α� 1) kt+1 + (1� α) nt+1 + At
�(82)
and log-linearization the equation
ct + kt+1 � (1� δ) kt = Atkαt n1�αt . (83)
yields
sc ct + si
�1δkt+1 �
(1� δ)
δkt
�= At + αkt + (1� α) nt (84)
and the technology follows
At = ρAt�1 + εt (85)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 31 / 46
Log linearizing the f.o.cs(continued):
Log linearization the equation
λt = βEt�λt+1
�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
��(81)
yields
λt = Et�
λt+1 + (1� β (1� δ))�(α� 1) kt+1 + (1� α) nt+1 + At
�(82)
and log-linearization the equation
ct + kt+1 � (1� δ) kt = Atkαt n1�αt . (83)
yields
sc ct + si
�1δkt+1 �
(1� δ)
δkt
�= At + αkt + (1� α) nt (84)
and the technology follows
At = ρAt�1 + εt (85)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 31 / 46
Log linearizing the f.o.cs(continued):
Log linearization the equation
λt = βEt�λt+1
�αAt+1kα�1
t+1 n1�αt+1 + 1� δ
��(81)
yields
λt = Et�
λt+1 + (1� β (1� δ))�(α� 1) kt+1 + (1� α) nt+1 + At
�(82)
and log-linearization the equation
ct + kt+1 � (1� δ) kt = Atkαt n1�αt . (83)
yields
sc ct + si
�1δkt+1 �
(1� δ)
δkt
�= At + αkt + (1� α) nt (84)
and the technology follows
At = ρAt�1 + εt (85)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 31 / 46
Reduced forms:
we can write consumption and labor as�ctnt
�= A1
�ktλt
�+ A2At (86)
and equation (82) and (84) become
Et
�kt+1λt+1
�= B
�ktλt
�+ R1Et At+1 + R2At . (87)
A unique equilibrium exists if one of the eigenvalues of B lies outsidethe unit circle and the other lies inside the unit circle.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 32 / 46
Reduced forms:
we can write consumption and labor as�ctnt
�= A1
�ktλt
�+ A2At (86)
and equation (82) and (84) become
Et
�kt+1λt+1
�= B
�ktλt
�+ R1Et At+1 + R2At . (87)
A unique equilibrium exists if one of the eigenvalues of B lies outsidethe unit circle and the other lies inside the unit circle.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 32 / 46
Reduced forms:
we can write consumption and labor as�ctnt
�= A1
�ktλt
�+ A2At (86)
and equation (82) and (84) become
Et
�kt+1λt+1
�= B
�ktλt
�+ R1Et At+1 + R2At . (87)
A unique equilibrium exists if one of the eigenvalues of B lies outsidethe unit circle and the other lies inside the unit circle.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 32 / 46
Solution Method 1:
(Remember to put back Et operator later)�kt+1λt+1
�= PΛP�1
�ktλt
�+ R1At+1 + R2At , (88)
which implies
P�1�kt+1λt+1
�= ΛP�1
�ktλt
�+ P�1R1At+1 + P�1R2At , (89)
or �x1t+1x2t+1
�=
�λ1 00 λ2
��x1tx2t
�+ R1At+1 + R2At . (90)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 33 / 46
Solution Method 1:
(Remember to put back Et operator later)�kt+1λt+1
�= PΛP�1
�ktλt
�+ R1At+1 + R2At , (88)
which implies
P�1�kt+1λt+1
�= ΛP�1
�ktλt
�+ P�1R1At+1 + P�1R2At , (89)
or �x1t+1x2t+1
�=
�λ1 00 λ2
��x1tx2t
�+ R1At+1 + R2At . (90)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 33 / 46
Solution Method 1:
(Remember to put back Et operator later)�kt+1λt+1
�= PΛP�1
�ktλt
�+ R1At+1 + R2At , (88)
which implies
P�1�kt+1λt+1
�= ΛP�1
�ktλt
�+ P�1R1At+1 + P�1R2At , (89)
or �x1t+1x2t+1
�=
�λ1 00 λ2
��x1tx2t
�+ R1At+1 + R2At . (90)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 33 / 46
Solution Method 1 (continued):
Suppose jλ2j > 1, then
Etx2t+1 = λ2xt + r1Et At+1 + r2At , (91)
orx2t =
1λ2Etx2t+1 �
1λ2
�r1Et At+1 + r2At
�.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 34 / 46
Solution Method 1 (continued):
Denote the foward operator as F ,
Fxt = Etxt+1,FFxt = FEtxt+1 = Et [Et+1xt+2] = Etxt+2 (92)
Iterating this forward and applying the law of iterated expectations, astationary solution for this di¤erence equation is given by
x2t =� 1
λ2
1� 1λ2F
�r1Et At+1 + r2At
= � r1
λ2
∞
∑j=0
�1
λ2
�jEt At+1+j �
r2λ2
∞
∑j=0
�1
λ2
�jEt At+j
= � r1λ2
ρ
1� ρλ2
At �r2λ2
11� ρ
λ2
At
= �ρr1 + r2λ2 � ρ
At . (93)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 35 / 46
Solution Method 1 (continued):
Denote the foward operator as F ,
Fxt = Etxt+1,FFxt = FEtxt+1 = Et [Et+1xt+2] = Etxt+2 (92)
Iterating this forward and applying the law of iterated expectations, astationary solution for this di¤erence equation is given by
x2t =� 1
λ2
1� 1λ2F
�r1Et At+1 + r2At
= � r1
λ2
∞
∑j=0
�1
λ2
�jEt At+1+j �
r2λ2
∞
∑j=0
�1
λ2
�jEt At+j
= � r1λ2
ρ
1� ρλ2
At �r2λ2
11� ρ
λ2
At
= �ρr1 + r2λ2 � ρ
At . (93)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 35 / 46
Solution Method 1 (continued):
since �x1tx2t
�� P�1
�ktλt
�, (94)
we havex2t = p21kt + p22λt .
Thus, the equilibrium path for λt is given by
λt = �p21p22kt �
1p22
ρr1 + r2λ2 � ρ
At . (95)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 36 / 46
Solution Method 1 (continued):
since �x1tx2t
�� P�1
�ktλt
�, (94)
we havex2t = p21kt + p22λt .
Thus, the equilibrium path for λt is given by
λt = �p21p22kt �
1p22
ρr1 + r2λ2 � ρ
At . (95)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 36 / 46
Solution Method 1 (continued):
Once we obtain λt , we have�ctnt
�= Π
�ktAt
�(96)
and
kt+1 = π
�ktAt
�. (97)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 37 / 46
Solution Method 1 (continued):
Once we obtain λt , we have�ctnt
�= Π
�ktAt
�(96)
and
kt+1 = π
�ktAt
�. (97)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 37 / 46
Solution Method 2
We have �ctnt
�= A2�3
0@ ktAtλt
1A (98)
and
Et
0@ kt+1At+1λt+1
1A = B3�3
0@ ktAtλt
1A = PΛP�1
0@ ktAtλt
1A . (99)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 38 / 46
Solution Method 2
We have �ctnt
�= A2�3
0@ ktAtλt
1A (98)
and
Et
0@ kt+1At+1λt+1
1A = B3�3
0@ ktAtλt
1A = PΛP�1
0@ ktAtλt
1A . (99)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 38 / 46
Solution Method 2 (continued)
De�ne
xt = P�1
0@ ktAtλt
1A , (100)
and suppose jλ3j > 1. Then a stationary solution is given by
x3t = 0, (101)
orp31kt + p32At + p33λt = 0, (102)
orλt = �
p31p33kt �
p32p33At . (103)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 39 / 46
Solution Method 2 (continued)
De�ne
xt = P�1
0@ ktAtλt
1A , (100)
and suppose jλ3j > 1. Then a stationary solution is given by
x3t = 0, (101)
orp31kt + p32At + p33λt = 0, (102)
orλt = �
p31p33kt �
p32p33At . (103)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 39 / 46
Solution Method 2 (continued)
De�ne
xt = P�1
0@ ktAtλt
1A , (100)
and suppose jλ3j > 1. Then a stationary solution is given by
x3t = 0, (101)
orp31kt + p32At + p33λt = 0, (102)
orλt = �
p31p33kt �
p32p33At . (103)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 39 / 46
Solution Method 2 (continued)
De�ne
xt = P�1
0@ ktAtλt
1A , (100)
and suppose jλ3j > 1. Then a stationary solution is given by
x3t = 0, (101)
orp31kt + p32At + p33λt = 0, (102)
orλt = �
p31p33kt �
p32p33At . (103)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 39 / 46
An example
Suppose we have �ctkt
�= QEt
�ct+1kt+1
�(104)
Where ct is the endogenous variable and kt is the state variable. Thematrix Q is:
Q =��2.5 �91.5 5
�(105)
Solve ct in term of kt .
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 40 / 46
An example
Suppose we have �ctkt
�= QEt
�ct+1kt+1
�(104)
Where ct is the endogenous variable and kt is the state variable. Thematrix Q is:
Q =��2.5 �91.5 5
�(105)
Solve ct in term of kt .
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 40 / 46
An example
Follow the procedure in class, The system can be written as :�ctkt
�=
��3 21 �1
���0.5 00 2
����1 �2�1 �3
� �ct+1kt+1
�(106)
Therefore we havect + 2kt = 0 (107)
orct = �2kt (108)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 41 / 46
An example
Follow the procedure in class, The system can be written as :�ctkt
�=
��3 21 �1
���0.5 00 2
����1 �2�1 �3
� �ct+1kt+1
�(106)
Therefore we havect + 2kt = 0 (107)
orct = �2kt (108)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 41 / 46
Calibration
There are 5 structural parameters in the model, fα, β, δ,γ, ρg ,so weneed �ve conditions
we can use the steady-state relationships to back solve the parametervalues so that the implied steady-state values of model are consistentwith empirical data.
In the steady-state
1 = β (1+ r �)
α =(r � + δ)k�
y �; 1� α =
wn�
y �
k�
y �=
αβ
1� β (1� δ).
If r = 4% a year (i.e., 1% a quarter), then in a quarterly model wehave β = 1
1.01 = 0.99.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 42 / 46
Calibration
There are 5 structural parameters in the model, fα, β, δ,γ, ρg ,so weneed �ve conditions
we can use the steady-state relationships to back solve the parametervalues so that the implied steady-state values of model are consistentwith empirical data.
In the steady-state
1 = β (1+ r �)
α =(r � + δ)k�
y �; 1� α =
wn�
y �
k�
y �=
αβ
1� β (1� δ).
If r = 4% a year (i.e., 1% a quarter), then in a quarterly model wehave β = 1
1.01 = 0.99.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 42 / 46
Calibration
There are 5 structural parameters in the model, fα, β, δ,γ, ρg ,so weneed �ve conditions
we can use the steady-state relationships to back solve the parametervalues so that the implied steady-state values of model are consistentwith empirical data.
In the steady-state
1 = β (1+ r �)
α =(r � + δ)k�
y �; 1� α =
wn�
y �
k�
y �=
αβ
1� β (1� δ).
If r = 4% a year (i.e., 1% a quarter), then in a quarterly model wehave β = 1
1.01 = 0.99.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 42 / 46
Calibration
There are 5 structural parameters in the model, fα, β, δ,γ, ρg ,so weneed �ve conditions
we can use the steady-state relationships to back solve the parametervalues so that the implied steady-state values of model are consistentwith empirical data.
In the steady-state
1 = β (1+ r �)
α =(r � + δ)k�
y �; 1� α =
wn�
y �
k�
y �=
αβ
1� β (1� δ).
If r = 4% a year (i.e., 1% a quarter), then in a quarterly model wehave β = 1
1.01 = 0.99.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 42 / 46
Calibration (continued)
wn�y � = 1� α; and wn�
y � = 0.65 in data, so we have α = 0.35.
k �y � = 10 in data, by α yk = (r + δ), we have
δ =α
k/y� r = 0.035� 0.01 = 0.025. (109)
implies a steady-state rate of saving equal to
s� = δky= 0.25 = 25%, (110)
and a steady-state cy = 1� s = 0.75.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 43 / 46
Calibration (continued)
wn�y � = 1� α; and wn�
y � = 0.65 in data, so we have α = 0.35.k �y � = 10 in data, by α yk = (r + δ), we have
δ =α
k/y� r = 0.035� 0.01 = 0.025. (109)
implies a steady-state rate of saving equal to
s� = δky= 0.25 = 25%, (110)
and a steady-state cy = 1� s = 0.75.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 43 / 46
Calibration (continued)
wn�y � = 1� α; and wn�
y � = 0.65 in data, so we have α = 0.35.k �y � = 10 in data, by α yk = (r + δ), we have
δ =α
k/y� r = 0.035� 0.01 = 0.025. (109)
implies a steady-state rate of saving equal to
s� = δky= 0.25 = 25%, (110)
and a steady-state cy = 1� s = 0.75.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 43 / 46
Calibration (continued)
Also, we know that the fraction of hours worked each week is aboutn� = 40
24�7 � . 24 (which implies that the fraction of hours worked ina quarter is also 0.24). Given the steady-state relationship,
γn1� n = (1� α)
yc=0.760.24
� 0.650.75
= 2.74, (111)
Finally, to calibrate ρ, we can estimate the Solow residual using
At = yt � αkt � (1� α) nt ,
and then estimate ρ by
At = ρAt�1 + et . (112)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 44 / 46
Calibration (continued)
Also, we know that the fraction of hours worked each week is aboutn� = 40
24�7 � . 24 (which implies that the fraction of hours worked ina quarter is also 0.24). Given the steady-state relationship,
γn1� n = (1� α)
yc=0.760.24
� 0.650.75
= 2.74, (111)
Finally, to calibrate ρ, we can estimate the Solow residual using
At = yt � αkt � (1� α) nt ,
and then estimate ρ by
At = ρAt�1 + et . (112)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 44 / 46
Simulation
The decisions rules can be arranged into:0BBB@ctntyt...
1CCCA = Π�ktAt
�(113)
�ktAt
�= M
�kt�1At�1
�+
�01
�εt . (114)
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 45 / 46
Simulation (continued)
Starting from the steady state at t = 0, we have k0�j = 0 andA0�j = 0 for j > 0. Given the sequence, fεtgTt=0 (drawn from arandom generator), (8) implies�
k0A0
�=
�01
�ε0 (115)
�k1A1
�= M
�k0A0
�+
�01
�ε1 (116)�
kt+1At+1
�= M
�kt�1At�1
�+
�01
�εt+1. (117)
Substituting the generated sequences,�kt , At
Tt=0 , into (113)
produces the sequences fct , nt , yt , ...gTt=0.
pfwang (Institute) Notes on the Real Business Cycle Model 03/09 46 / 46