Stochastic Growth Model

KoenVermeylen
TheStochasticGrowthModel
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The Stochastic Growth Model

2

Contents
1. Introduction
2. The stochastic growth model
3. The steady state
4. Linearization around the balanced growth path
5. Solution of the linearized model
6. Impulse response functions
7. Conclusions
Appendix A
A1. The maximization problem of the representative fi rm
A2. The maximization problem of the representative household
Appendix B
Appendix C
C1. The linearized production function
C2. The linearized law of motion of the capital stock
C3. The linearized fi rst-order condotion for the fi rm’s labor demand
C4. The linearized fi rst-order condotion for the fi rm’s capital demand
C5. The linearized Euler equation of the representative household
C6. The linearized equillibrium condition in the goods market
References
Contents

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The Stochastic Growth Model

3

Introduction
1. Introduction
This article present s the stoch a stic growth model. The stochastic growth model
is a stochastic version of the neoclassical growth model with microfoundations,
1
and provides the backbone of a lot of macroeconomic models that are used in
modern macroeconomic research. The most popular way t o solve the stochastic
growth model, is to l inearize the model around a steady state,
2
and to solve the
linearized model with the method of undetermined coefficients . This solution
method is due to Campbell (1994).
The set-up of the stochastic growth mode l is given in the next section. Section 3
solv e s for the steady s tate, around w hich the model is linearized in section 4. The
linearized model is then solved in section 5. Section 6 show s how the economy
responds to stochastic shocks. Some c oncluding remarks are given i n section 7.
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The Stochastic Growth Model

4

The representative firm Assume that the production side of the economy
is represented by a representative firm, which produces output ac c ording to a
Cobb-Douglas production function:
Y
t
= K
α
t
(A
t
L
t
)
1−α
with 0 <α<1(1)
Y is aggregate output, K is the aggregate capital stock, L is aggregate labor
supply and A is a technology parameter. The subscript t denotes the time period.
The aggregate capital stock depends on aggregate investment I and the depreci-
ation rate δ:
K
t+1
=(1− δ)K
t
+ I
t
with 0 ≤ δ ≤ 1(2)
2. The stochastic growth model
The productivity parameter A follows a stochastic path with trend growth g and

an AR(1) stochastic component:
ln A
t
=lnA
∗
t
+
ˆ
A
t
ˆ
A
t
= φ
A
ˆ
A
t−1
+ ε
A,t
with |φ
A
| < 1(3)
A
∗
t
= A
∗
t−1
(1 + g)

The stoch astic shock ε
A,t
is i.i.d. with mean zero.
The goods market alway s clears, such that the firm alway s sells i ts total pro-
duction. Taking current and future factor prices as given, the firm hires labor
and invests in its capital stock to maximize its current value. This leads to the
following first-order-conditions:
3
(1 − α)
Y
t
L
t
= w
t
(4)
1=E
t

1
1+r
t+1
α
Y
t+1
K
t+1

+ E
t


1 − δ
1+r
t+1

(5)
According to equation (4), the firm hires labor until the marginal product of
labor is equal to its m arginal c ost (whic h is the real wage w). Equation (5) s hows
that the firm’s investment demand at time t is such that the marginal cost of
in vestment, 1, is equal to the expected discoun te d marginal product o f capital at
time t + 1 plus the expected discounted value of the extra capital stock which is
left after depreciation at time t +1.
The stochastic growth model
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The Stochastic Growth Model

5

The government The government consumes every period t an amount G
t
,
which follow s a stochastic path with trend growth g and an AR(1) stochastic
component:
ln G
t
=lnG
∗
t
+

ˆ
G
t
ˆ
G
t
= φ
G
ˆ
G
t−1
+ ε
G,t
with |φ
G
| < 1(6)
G
∗
t
= G
∗
t−1
(1 + g)
The stochastic shock ε
G,t
is i.i.d. with mean zero. ε
A
and ε
G
are uncorrelated

at all leads and lags. The government finances its consumption by issuing public
debt, subject to a transversality condition,
4
and by raising lump-sum taxes.
5
The
timing of taxation is irrelevant because of Ricardian Equivalence.
6
The stochastic growth model
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The Stochastic Growth Model

6

The representative household There is one represen tative household, who
derives utility from her current and future consumption:
U
t
= E
t


∞

s=t

1
1+ρ

s−t
ln C
s

with ρ>0(7)
The parameter ρ is c alled the s ubjective discount rate.
Every period s, the household starts off with her assets X
s
and receives interest
payments X
s
r
s
. She also supplies L units of labor to the representative firm, and
therefore receives labor income w
s
L. Tax pa yments are lump-sum and amount to
T
s
. She then decides how much she consumes, and how much assets she will hold
in her portfolio unt il period s + 1. This leads to her dynamic budget constraint:
X

s+1
= X
s
(1 + r
s
)+w
s
L − T
s
− C
s
(8)
We need to make sure that the household does not incur ever increasing debts,
which she will never be able to pay back anymore. Under plausible assumptions,
this implies that over an infinitely long horizon the present discounted value of
the household’s assets must be zero:
lim
s→∞
E
t

s

s
′
=t
1
1+r
s
′


X
s+1

=0 (9)
This equation is called the transversality condition.
The household then takes X
t
and the current and e xpected values of r, w,andT
as giv e n, and chooses her consumption path to maximize her utilit y (7) subject
to her dynamic budget constraint (8) and the transversalit y condition (9). This
leads to the following Euler equation:
7
1
C
s
= E
s

1+r
s+1
1+ρ
1
C
s+1

(10)
Equilibrium Every period, the factor markets and the goods market clear. For
the labor market, we already implicitly assumed this by using the same notation
(L) for the representative h ousehold’s labor supply and the representative firm’s

labor demand. Equilibrium in the goods market requires that
Y
t
= C
t
+ I
t
+ G
t
(11)
Equilibrium in the capital market follows then from Walras’ law.
The stochastic growth model
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The Stochastic Growth Model

7

Let us now d erive the model’s balanced growth path (or steady state); variables
evaluated on the balanced growth path are denoted by a
∗
.
To derive the balanced gro wth path, we assume that b y sheer luc k ε
A,t
=
ˆ
A
t
=
ε

G,t
=
ˆ
G
t
=0,∀t. The model then becomes a standard neo classical growth
model, for w hich the solution is given by:
8
Y
∗
t
=

α
r
∗
+ δ

α
1−α
A
∗
t
L (12)
K
∗
t
=

α

r
∗
+ δ

1
1−α
A
∗
t
L (13)
I
∗
t
=(g + δ)

α
r
∗
+ δ

1
1−α
A
∗
t
L (14)
C
∗
t
=


1 − (g + δ)
α
r
∗
+ δ

α
r
∗
+ δ

α
1−α
A
∗
t
L − G
∗
t
(15)
w
∗
t
=(1− α)

α
r
∗
+ δ


α
1−α
A
∗
t
(16)
r
∗
=(1+ρ)(1 + g) − 1 (17)
3. The steady state
The steady state
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The Stochastic Growth Model

8

Let us now linearize t he model presented in section 2 around the balance d grow th
path derived in section 3. Loglinear deviations from the balanced growth path
are denoted b y aˆ(so that
ˆ
X =lnX − ln X
∗
).
Below a re the loglinearized versions of the production function ( 1), the law of
motion of the capital stock (2), the first-order conditions (4) and (5), the Euler
equation (10) and the equilibrium condition ( 11):
9
ˆ
Y
t
= α
ˆ
K
t
+(1− α)
ˆ
A
t
(18)
ˆ
K
t+1
=

1 − δ
1+g
ˆ
K
t
+
g + δ
1+g
ˆ
I
t
(19)
ˆ
Y
t
=ˆw
t
(20)
E
t

r
t+1
− r
∗
1+r
∗

=
r

∗
+ δ
1+r
∗

E
t
(
ˆ
Y
t+1
) − E
t
(
ˆ
K
t+1
)

(21)
4. Linearization around the balanced growth path
ˆ
C
t
= E
t

ˆ
C
t+1


− E
t

r
t+1
− r
∗
1+r
∗

(22)
ˆ
Y
t
=
C
∗
t
Y
∗
t
ˆ
C
t
+
I
∗
t
Y

∗
t
ˆ
I
t
+
G
∗
t
Y
∗
t
ˆ
G
t
(23)
The loglinearized la ws of motion of A and G are given by equations (3) and (6):
ˆ
A
t+1
= φ
A
ˆ
A
t
+ ε
A,t+1
(24)
ˆ
G

t+1
= φ
G
ˆ
G
t
+ ε
G,t+1
(25)
Linearization around the balanced growth path
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The Stochastic Growth Model

9

I now solve the linearized model, which is described by equations (18) until (25).
First note that
ˆ
K
t
,
ˆ
A
t
and
ˆ
G
t
are known in the beginning of period t:
ˆ

K
t
dep ends
on past investment decisions, and
ˆ
A
t
and
ˆ
G
t
are determined by current and past
values of respectively ε
A
and ε
G
(which are exogenous).
ˆ
K
t
,
ˆ
A
t
and
ˆ
G
t
are
therefore called period t’s state variables. The values of the other variables in

period t are endogenous, however: investment and consumption are chosen by
the representative firm and the representative household in such a way that they
maximize their profits and utility (
ˆ
I
t
and
ˆ
C
t
are therefore called period t’s control
variables); the values of the interest rate and the wage are such that they clear
the capital and the labor marke t.
Solving the model requires that we express period t’s endogenous variables as
functions of period t’s state variables. The solution of
ˆ
C
t
, f or instance, therefore
looks as follows:
ˆ
C
t
= ϕ
CK
ˆ
K
t
+ ϕ
CA

ˆ
A
t
+ ϕ
CG
ˆ
G
t
(26)
The cha llenge now is to determine the ϕ-coefficients.
First substitute equation (26) in t he Euler equation (22):
ϕ
CK
ˆ
K
t
+ ϕ
CA
ˆ
A
t
+ ϕ
CG
ˆ
G
t
= E
t

ϕ

CK
ˆ
K
t+1
+ ϕ
CA
ˆ
A
t+1
+ ϕ
CG
ˆ
G
t+1

− E
t

r
t+1
− r
∗
1+r
∗

(27)
Now eliminate E
t
[(r
t+1

−r
∗
)/(1+r
∗
)] with equation (21), and use equations (18),
(24) and (25) to eliminate
ˆ
Y
t+1
,
ˆ
A
t+1
and
ˆ
G
t+1
in the resulting expression. This
5. Solution of the linearized model
Solution of the linearized model
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The Stochastic Growth Model

10

leads to a relation between period t’s state variables, the ϕ-coefficients and
ˆ
K
t+1
:

ϕ
CK
ˆ
K
t
+ ϕ
CA
ˆ
A
t
+ ϕ
CG
ˆ
G
t
=

ϕ
CK
+(1− α)
r
∗
+ δ
1+r
∗

ˆ
K
t+1
+


ϕ
CA
− (1 − α)
r
∗
+ δ
1+r
∗

φ
A
ˆ
A
t
+ ϕ
CG
φ
G
ˆ
G
t
(28)
We now derive a second r elation be tween period t’s state variables, the ϕ-coe fficients
and
ˆ
K
t+1
: rewrite the law of motion (19) by eliminating
ˆ

I
t
with equation (23);
eliminate
ˆ
Y
t
and
ˆ
C
t
in the resulting equation with the production function (18)
and expression (26); note that I
∗
= K
∗
(g + δ); and note that (1 − δ)/(1 + g)+
(αY
∗
t
)/(K
∗
t
(1 + g)) = (1 + r
∗
)/(1 + g). This yields:
ˆ
K
t+1
=


1+r
∗
1+g
−
C
∗
K
∗
(1 + g)
ϕ
CK

ˆ
K
t
+

(1 − α)Y
∗
K
∗
(1 + g)
−
C
∗
K
∗
(1 + g)
ϕ

CA

ˆ
A
t
−

G
∗
K
∗
(1 + g)
+
C
∗
K
∗
(1 + g)
ϕ
CG

ˆ
G
t
(29)
Substituting equation (29) in equation (28) to eliminate
ˆ
K
t+1
yields:

ϕ
CK
ˆ
K
t
+ ϕ
CA
ˆ
A
t
+ ϕ
CG
ˆ
G
t
=

ϕ
CK
+(1− α)
r
∗
+ δ
1+r
∗

1+r
∗
1+g
−

C
∗
K
∗
(1 + g)
ϕ
CK

ˆ
K
t
+

ϕ
CK
+(1− α)
r
∗
+ δ
1+r
∗

(1 − α)Y
∗
K
∗
(1 + g)
−
C
∗

K
∗
(1 + g)
ϕ
CA

ˆ
A
t
−

ϕ
CK
+(1− α)
r
∗
+ δ
1+r
∗

G
∗
K
∗
(1 + g)
+
C
∗
K
∗

(1 + g)
ϕ
CG

ˆ
G
t
+

ϕ
CA
− (1 − α)
r
∗
+ δ
1+r
∗

φ
A
ˆ
A
t
− ϕ
CG
φ
G
ˆ
G
t

(30)
As this equation must hold for all values of
ˆ
K
t
,
ˆ
A
t
and
ˆ
G
t
, we find the following
system of three equations and three unknowns:
ϕ
CK
=

ϕ
CK
+(1− α)
r
∗
+ δ
1+r
∗

1+r
∗

1+g
−
C
∗
K
∗
(1 + g)
ϕ
CK

(31)
ϕ
CA
=

ϕ
CK
+(1− α)
r
∗
+ δ
1+r
∗

(1 − α)Y
∗
K
∗
(1 + g)
−

C
∗
K
∗
(1 + g)
ϕ
CA

+

ϕ
CA
− (1 − α)
r
∗
+ δ
1+r
∗

φ
A
(32)
ϕ
CG
= −

ϕ
CK
+(1− α)
r

∗
+ δ
1+r
∗

G
∗
K
∗
(1 + g)
+
C
∗
K
∗
(1 + g)
ϕ
CG

− ϕ
CG
φ
G
(33)
Solution of the linearized model
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The Stochastic Growth Model

11

Now note that equation (31) is quadratic in ϕ
CK
:
Q
0
+ Q
1
ϕ
CK
+ Q
2
ϕ
2
CK
= 0 (34)
where Q
0
= −(1 − α)
r
∗
+δ
1+g
, Q
1
=(1− α)
r
∗
+δ
1+r

∗
C
∗
t
K
∗
t
(1+g)
−
r
∗
−g
1+g
and Q
2
=
C
∗
t
K
∗
t
(1+g)
This quadratic equation has two solutions:
ϕ
CK1,2
=
−Q
1
±


Q
2
1
− 4Q
0
Q
2
2Q
2
(35)
It turns o ut t hat one of these t wo solutions yields a stable dynamic system, while
the other one yields an unstable dynamic system. This can be recognized as
follows.
Recall that there are three state variables in this economy: K, A and G. A
and G may undergo shocks that pull them away from their steady states, but
as |φ
A
| and |φ
G
| are less than one, equations (3) and (6) imply that they are
always expected to converge back to their steady state values. Let us now look
at the expected time path f or K, which is described by equation ( 29). I f K is not
at its steady state value (i.e. if
ˆ
K =0),K is expected to converge back to its
steady state value if the absolute value of the coefficient of
ˆ
K
t

in equation (29),
1+r
∗
1+g
−
C
∗
K
∗
(1+g)
ϕ
CK
,islessthanone;if|
1+r
∗
1+g
−
C
∗
K
∗
(1+g)
ϕ
CK
| > 1,
ˆ
K i s expected to
increase - which means that K is expected to run away along an explosive path,
ever further a way from its steady state.
Solution of the linearized model

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The Stochastic Growth Model

12

ϕ
CG
= −

ϕ
CK
+(1− α)
r
∗
+δ
1+r
∗

G
∗
K
∗
(1+g)

1+

ϕ
CK
+(1− α)
r
∗
+δ
1+r
∗

C
∗
K
∗
(1+g)
− φ
G
(38)
We now have found all the ϕ-coefficients o f equation (26), so we can compute
ˆ
C
t
from period t’s state variables
ˆ
K
t
,
ˆ
A

t
and
ˆ
G
t
.Onceweknow
ˆ
C
t
, the other
endogenous variables can easily be found from equations (18), (19), (20), (21)
and (23). The values of the state variables in period t + 1 can be computed from
equation (29), and equations (3) and (6) (moved one period forward).
Now note that equation (31) is quadratic in ϕ
CK
:
Q
0
+ Q
1
ϕ
CK
+ Q
2
ϕ
2
CK
= 0 (34)
where Q
0

= −(1 − α)
r
∗
+δ
1+g
, Q
1
=(1− α)
r
∗
+δ
1+r
∗
C
∗
t
K
∗
t
(1+g)
−
r
∗
−g
1+g
and Q
2
=
C
∗

t
K
∗
t
(1+g)
This quadratic equation has two solutions:
ϕ
CK1,2
=
−Q
1
±

Q
2
1
− 4Q
0
Q
2
2Q
2
(35)
It turns o ut t hat one of these t wo solutions yields a stable dynamic system, while
the other one yields an unstable dynamic system. This can be recognized as
follows.
Recall that there are three state variables in this economy: K, A and G. A
and G may undergo shocks that pull them away from their steady states, but
as |φ
A

| and |φ
G
| are less than one, equations (3) and (6) imply that they are
always expected to converge back to their steady state values. Let us now look
at the expected time path f or K, which is described by equation ( 29). I f K is not
at its steady state value (i.e. if
ˆ
K =0),K is expected to converge back to its
steady state value if the absolute value of the coefficient of
ˆ
K
t
in equation (29),
1+r
∗
1+g
−
C
∗
K
∗
(1+g)
ϕ
CK
,islessthanone;if|
1+r
∗
1+g
−
C

∗
K
∗
(1+g)
ϕ
CK
| > 1,
ˆ
K i s expected to
increase - which means that K is expected to run away along an explosive path,
ever further a way from its steady state.
Solution of the linearized model
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The Stochastic Growth Model

13

We now calibrate the model by assigning appropriate values to α, δ, ρ, A
∗
t
, G
∗
t
,
φ
A
, φ
G
, g and L. Let us assume, for instance, t hat every period corresponds
to a quarter, and let us choose parameter values that mimic the U.S. economy:

α =1/3, δ =2.5%, φ
A
=0.5, φ
G
=0.5, and g =0.5%; A
∗
t
and L are normalized
to 1; G
∗
t
is chosen such that G
∗
t
/Y
∗
t
= 20%; and ρ is chosen such that r
∗
=1.5%.
11
It is then straightforward to compute the balanced growth path: Y
∗
t
=2.9,
K
∗
t
=24.1, I
∗

t
=0.7, C
∗
t
=1.6andw
∗
t
=1.9 (while r
∗
=1.5% per construction).
Y
∗
, K
∗
, I
∗
, C
∗
and w
∗
all grow at rate 0.5% per quarter, while r
∗
remains
constant ove r time. Note that this parameterization yields an annual capital-
output-ratio of about 2, while C and I are about 55% and 25% of Y , respectively
- whic h seem reasonable numbers. Once we have computed the steady state, w e
can use equations (36), (37) and (38) to compute the ϕ-coefficients. We are the n
ready to trace out the e conomy’s reaction to shocks in A and G.
Consider first t he effect of a technology shock in quarter 1. Suppose the econom y
is initially moving along its balanced growth path (such that

ˆ
K
s
=
ˆ
A
s
=
ˆ
G
s
=0
∀s<1), when in quarter 1 it is suddenly hit by a technology shock ε
A,1
=1.
From equation (3) follows then that
ˆ
A
1
= 1 as well, while equations (29) and
(6) imply that
ˆ
K
1
=
ˆ
G
1
= 0. Given these values for quarter 1’s state variables
and given the ϕ-coefficients,

ˆ
C
1
can be computed from equation (26); the other
endogenous va riables in quarter 1 follow from equations (18), (19), (20), (21)
and (23). Quarter 2’s state va riables can then be computed from equations (28),
(3) and (6) - which leads to the values for quarter 2’s endogenous variables,
and quarter 3’s state variables. In this way, we can trace out the effect of the
technology sho ck into the infinite future.
6. Impulse response functions
Impulse response functions
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The Stochastic Growth Model

14

Figure 1: Effect of a 1% shock in A
0
0.05
0.10
0.15
0 4 8 1216202428323640
in %
quarter
on
ˆ
K
0
0.2
0.4

0.6
0.8
0 4 8 1216202428323640
in %
quarter
on
ˆ
Y
0
0.05
0.10
0.15
0 4 8 1216202428323640
in %
quarter
on
ˆ
C
0
1
2
3
0 4 8 1216202428323640
in %
quarter
on
ˆ
I
0
0.2

0.4
0.6
0.8
0 4 8 1216202428323640
in %
quarter
on ˆw
−0.5
0
0.5
1.0
1.5
0 4 8 1216202428323640
in %
quarter
on E(r) − r
∗
Impulse response functions
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The Stochastic Growth Model

15

Figure 2: Effect of a 1% shock in G
−0.04
−0.03
−0.02
−0.01
0
0 4 8 1216202428323640

in %
quarter
on
ˆ
K
−0.015
−0.010
−0.005
0
0 4 8 1216202428323640
in %
quarter
on
ˆ
Y
−0.04
−0.03
−0.02
−0.01
0
0 4 8 1216202428323640
in %
quarter
on
ˆ
C
−0.8
−0.6
−0.4
−0.2

0
0 4 8 1216202428323640
in %
quarter
on
ˆ
I
−0.015
−0.010
−0.005
0
0 4 8 1216202428323640
in %
quarter
on ˆw
0
0.04
0.08
0.12
0 4 8 1216202428323640
in %
quarter
on E(r) − r
∗
Impulse response functions
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The Stochastic Growth Model

16

Figure 1 shows how the economy reacts during the first 40 quarters. Note that
Y jumps up in quarter 1, together with the technology shock. As a result, the
representative household increases her consumption, but as she w ant s to smooth
her consumption over time, C increases less than Y . Investment I therefore
initially increases more than Y .AsI increases, the capital stock K gradually
increases as well after period 1. The expected rate of return, E(r), is at first higher
than on the balanced growth path (thanks to the technology shock). However,
as the technology shock dies out w hile t he capital stock builds up, the expected
interest rate rapidly falls and even becomes negative after a few quarters. The real
w age w follows the time path of Y . Note that all variables eventually converge
back to their steady state values.
Consider now the effect of a shock in government exp enditures in quarter 1.
Assume again that the economy is on a balanced gro wth path in quarter 0. I n
quarter 1, ho wever, the economy is hit by a shock in government expenditures
ε
G,1
= 1. From equation (3) follows then that
ˆ
A
1
= 1 as well, while equations
(29) and (6) imply that
ˆ
K
1
=
ˆ
G
1

= 0. Once we know the state variables in
quarter 1, we can compute the endogenous variables in quarter 1 and the state
variables for quarter 2 in the same way as in t he case of a technology shock -
which leads to the values for quarter 2’s endogenous variables and quarter 3’s
state variables, and so on until the infinite future.
Impulse response functions
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The Stochastic Growth Model

17

Figure 2 shows the e conomy’s reaction to a shock in governmen t expenditures
during the first 40 quarters. As G i ncreases, E(r) increases as well suc h that C
and I fall (to make sure that C+I +G remains equal to Y , which does not change
in quarter 1 as
ˆ
K
1
=0). AsI falls, the capital stock K gradually decreases after
period 1, such that Y starts decreasing after period 1 as well. In the meantime,
however, the shock in G isdyingout,soafterawhileE(r) dec reases again. As a
result, C and I recover - and as I recovers, K and Y recover also. Note that the
real w age w again follows the time path of Y . Eventually, all variables converge
back to their steady state values.
Impulse response functions
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The Stochastic Growth Model

18

This note presented the sto chastic growth mod el, and solved the model by first
linearizing i t around a steady state and by then solving the linearized model with
the method of undetermined coefficients.
Ev en though the stochastic growth model itself might bear little resemblance to
7. Conclusions
the real world, it has proven to be a useful framewo rk that can easily be extended
to accoun t for a wide range of macroeconomic issues that are potentially impor-
tan t. Kydland and Prescott (1982) introduced labor/leisure-substitution in the
stochastic growth model, which gave rise to the so-called real-business- cycle liter-
ature. Greenwood and Huffman (1991) and Baxter and King (1993) replaced the
lump-sum taxation by distortionary taxation, to study how taxes affect the be-
ha vior of firms and households. In the beginning of the 1990s, researc hers started
in troducing money and nominal rigidities in the model, which gave rise to New
Keynesian stochastic dynamic general equilibrium models that are now widely
used to study monetary policy - se e Goodfriend an d King (1997) for an overview.
Vermeylen (2006) shows how the representative household can be replaced by a
large number of households to study the effect of job insecurity on consumption
and saving in a ge neral equilibrium setting.
Conclusions
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The Stochastic Growth Model

19

1
Microfoundations means that the objectives of the economic agents are formulated ex-
plicitly, and that their behavior is derived by assuming that they always try t o achieve
their objectives as well as they can.
2
A steady state is a condition in which a num ber of key variables are not changing. In the
stochastic growth m odel, these key v ari ables are for instance the growth rate of aggregate
production, the interest rate and the capital-output-ratio.
3
See app endix A for derivations.
4
This means that the present discounted value of public debt in the distant future should
b e equal to zero, such that public debt cannot keep on rising at a rate that is higher
than the interest rate. This guarantees that public debt is always equal to the present
discounted value of the government’s future primary surpluses.
5
Lump-sum taxes do not affect the first-order conditions of the firms and the households,
and therefore do not affect their behavior either.
6

Ricardian equivalence is the phenomenon that - given certain assumptions - it turns out
to be irrelevant whether the government finances its expenditures by issuing public debt
or by raising taxes. The reason for this is that given the time path of go vernment expen-
ditures, every increase in public debt must so oner or later be matched by an increase in
taxes, such that the presen t discounted value of the taxes which a representative house-
hold has to pay is not affected by the way how the government finances its expenditures -
which implies t hat her current wealth and her consumption path are not affected either.
7
See appendix A for the derivation.
8
See appendix B for the derivation.
9
See appendix C for the derivations.
10
The solution with unstable dynamics not only does not make sense from an economic
point of view, it also violates the transversalit y conditions.
11
Note that these values imply that the annual depreciation rate, the annual growth rate
and the annual interest rate are about 10%, 2% and 6%, respectively.
Conclusions
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The Stochastic Growth Model

20

The maximization problem of the firm can be rewritten as:
V
t
(K
t

)= max
{L
t
,I
t
}

Y
t
− w
t
L
t
− I
t
+ E
t

1
1+r
t+1
V
t+1
(K
t+1
)

(A.1)
s.t. Y
t

= K
α
t
(A
t
L
t
)
1−α
K
t+1
=(1− δ)K
t
+ I
t
The first-order conditions for L
t
, respectively I
t
,are:
(1 − α)K
α
t
A
1−α
t
L
−α
t
− w

t
=0 (A.2)
−1+E
t

1
1+r
t+1
∂V
t+1
(K
t+1
)
∂K
t+1

=0 (A.3)
In addition, the envelope theorem implies that
∂V
t
(K
t
)
∂K
t
= αK
α−1
t
(A
t

L
t
)
1−α
+ E
t

1
1+r
t+1
∂V
t+1
(K
t+1
)
∂K
t+1

(1 − δ)(A.4)
Substituting the production function in (A.2) gives equation (4):
(1 − α)
Y
t
L
t
= w
t
Substituting (A.3) in (A.4) yields:
∂V
t

(K
t
)
∂K
t
= αK
α−1
t
(A
t
L
t
)
1−α
+(1− δ)
Moving one period forward, and substituting again in (A.3) gives:
−1+E
t

1
1+r
t+1

αK
α−1
t+1
(A
t+1
L
t+1

)
1−α
+(1− δ)


=0
Substituting the production function in the equation above and reshuffling leads to equa-
tion (5):
1=E
t

1
1+r
t+1
α
Y
t+1
K
t+1

+ E
t

1 − δ
1+r
t+1

A2. The maximization problem of the representative household
The maximization problem of the household can be rewritten as:
U

t
(X
t
)=max
{C
t
}

ln C
t
+
1
1+ρ
E
t
[U
t+1
(X
t+1
)]

(A.5)
s.t. X
t+1
= X
t
(1 + r
t
)+w
t

L − T
t
− C
t
Appendix A
A1. The maximization problem of the representative fi rm
A2. The maximization problem of the representative household
Appendix A
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The Stochastic Growth Model

21

The first-order condition for C
t
is:
1
C
t
−
1
1+ρ
E
t

∂U
t+1
(X
t+1

)
∂X
t+1

=0 (A.6)
In addition, the envelope theorem implies that
∂U
t
(X
t
)
∂X
t
=
1
1+ρ
E
t

∂U
t+1
(X
t+1
)
∂X
t+1
(1 + r
t
)


(A.7)
Substituting (A.6) in (A.7) yields:
∂U
t
(X
t
)
∂X
t
=(1+r
t
)
1
C
t
Moving one period forward, and substituting again in (A.6) gives the Euler equation
(10):
1
C
t
− E
t

1+r
t+1
1+ρ
1
C
t+1


=0
Appendix A
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thinking
.
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The Stochastic Growth Model

22

If C grows at rate g, the Euler equation (10) implies that
C
∗
s
(1 + g)=
1+r
∗
1+ρ
C
∗
s
Rearranging gives then the gross real rate of return 1 + r
∗
:
1+r
∗
=(1+g)(1 + ρ )
which immediately leads to equation (17).

Subsituting in the firm’s first-order condition (5) gives:
α
Y
∗
t+1
K
∗
t+1
= r
∗
+ δ
Using the production function (1) to eliminate Y yields:
αK
∗α−1
t+1
(A
t+1
L)
1−α
= r
∗
+ δ
Rearranging gives then the value of K
∗
t+1
:
K
∗
t+1
=


α
r
∗
+ δ

1
1−α
A
t+1
L
which is equivalent to equation (13).
Appendix B
Appendix B
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The Stochastic Growth Model

23

Substituting in the production function (1) gives then equation (12):
Y
∗
t
=

α
r
∗
+ δ


α
1−α
A
t
L
Substituting (12) in the first-order condition (4) gives equation (16):
w
∗
t
=(1− α)

α
r
∗
+ δ

α
1−α
A
t
Substituting (13) in the law of motion (2) yields:

α
r
∗
+ δ

1
1−α
A

t+1
L =(1− δ)

α
r
∗
+ δ

1
1−α
A
t
L + I
∗
t
such that I
∗
t
is given by:
I
∗
t
=

α
r
∗
+ δ

1

1−α
A
t+1
L − (1 − δ)

α
r
∗
+ δ

1
1−α
A
t
L
=

α
r
∗
+ δ

1
1−α
[(1 + g) − (1 − δ)] A
t
L
=(g + δ)

α

r
∗
+ δ

1
1−α
A
t
L
which is equation (14).
Consumption C
∗
can then be computed from the equilibrium condition in the goods
market:
C
∗
t
= Y
∗
t
− I
∗
t
− G
∗
t
=

α
r

∗
+ δ

α
1−α
A
t
L − (g + δ)

α
r
∗
+ δ

1
1−α
A
t
L − G
∗
t
=

1 − α
g + δ
r
∗
+ δ

α

r
∗
+ δ

α
1−α
A
t
L − G
∗
t
Now r ecall that on the balanced growth path, A and G grow at the rate of technological
progress g. The equation above then implies that C
∗
also grows at the rate g,suchthat
our initial educated guess turns out to be correct.
Appendix B
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The Stochastic Growth Model

24

The pro duction function is given by equation (1):
Y
t
= K
α
t
(A

t
L
t
)
1−α
Appendix C
C1. The linearized production function
Taking logarithms of both sides of this equation, and subtracting from both sides their
values on the balanced growth path (taking into account that
ˆ
L
t
= 0), immediately yields
the linearized version of the production function:
ln Y
t
= α ln K
t
+(1− α)lnA
t
+(1− α)lnL
t
ln Y
t
− ln Y
∗
t
= α(ln K
t
− ln K

∗
t
)+(1− α)(ln A
t
− ln A
∗
t
)+(1− α)(ln L
t
− ln L
∗
t
)
ˆ
Y
t
= α
ˆ
K
t
+(1− α)
ˆ
A
t
which is equation (18).
Appendix C
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The Stochastic Growth Model

25

The law of motion of the capital stock is given by equation (2):
K
t+1
=(1− δ)K
t
+ I
t
Taking logarithms of both sides of this equation, and subtracting from both sides their
values on the balanced growth path, yields:
ln K
t+1
− ln K
∗
t+1
=ln{(1 − δ)K
t
+ I
t
}−ln K
∗
t+1
Now take a first-order Taylor-approximation of the right-hand-side around ln K
t
=lnK
∗
t

and ln I
t
=lnI
∗
t
:
ln K
t+1
− ln K
∗
t+1
= ϕ
1
(ln K
t
− ln K
∗
t
)+ϕ
2
(ln I
t
− ln I
∗
t
)
ˆ
K
t+1
= ϕ

1
ˆ
K
t
+ ϕ
2
ˆ
I
t
(C.1)
where
ϕ
1
=

∂ ln {(1 − δ)K
t
+ I
t
}
∂ ln K
t

∗
ϕ
2
=

∂ ln {(1 − δ)K
t

+ I
t
}
∂ ln I
t

∗
ϕ
1
and ϕ
2
can be worked out as follows:
ϕ
1
=

∂ ln {(1 − δ)K
t
+ I
t
}
∂K
t
∂K
t
∂ ln K
t

∗
=


1 − δ
(1 − δ)K
t
+ I
t
K
t

∗
=

1 − δ
K
t+1
K
t

∗
=
1 − δ
1+g
as K
t
grows at rate g on the balanced growth path
ϕ
2
=

∂ ln {(1 − δ)K

t
+ I
t
}
∂I
t
∂I
t
∂ ln I
t

∗
C2. The linearized law of motion of the capital stock
=

1
(1 − δ)K
t
+ I
t
I
t

∗
=

1
K
t+1
I

t

∗
=
g + δ
1+g
as I
∗
t
/K
∗
t
= g + δ and K
t
grows at rate g on the balanced growth path
Substituting in equation (C.1) gives then the linearized law of motion for K:
ˆ
K
t+1
=
1 − δ
1+g
ˆ
K
t
+
g + δ
1+g
ˆ
I

t
which is equation (19).
Appendix C