The Stochastic Growth Model

32 Pages·1.092 MB·English

by  Vermeylen Koen.

The Stochastic Growth Model 
Koen Vermeylen 
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The Stochastic Growth Model Contents
Contents
1. Introduction 3
2. The stochastic growth model 4
3. The steady state 7
4. Linearization around the balanced growth path 8
5. Solution of the linearized model 9
6. Impulse response functions 13
7. Conclusions 18
Appendix A 20
A1. The maximization problem of the representative fi rm 20
A2. The maximization problem of the representative household 20
Appendix B 22
Appendix C 24
C1. The linearized production function 24
C2. The linearized law of motion of the capital stock 25
C3. The linearized fi rst-order condotion for the fi rm’s labor demand 26
C4. The linearized fi rst-order condotion for the fi rm’s capital demand 26
C5. The linearized Euler equation of the representative household 28
C6. The linearized equillibrium condition in the goods market 30
References 32
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The Stochastic Growth Model Introduction
1. Introduction
This article presents the stochastic growth model. The stochastic growth model
1
is a stochastic version of the neoclassical growth model with microfoundations,
and provides the backbone of a lot of macroeconomic models that are used in
modern macroeconomic research. The most popular way to solve the stochastic
2
growth model, is to linearize the model around a steady state, 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 model is given in the next section. Section 3
solves for the steady state, around which the model is linearized in section 4. The
linearized model is then solved in section 5. Section 6 shows how the economy
responds to stochastic shocks. Some concluding remarks are given in section 7.
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The Stochastic Growth Model The stochastic growth model
2. The stochastic growth model
The representative firm Assume that the production side of the economy
is represented by a representative firm, which produces output according to a
Cobb-Douglas production function:
Y = Kα(A L )1−α with 0 < α < 1 (1)
t t t t
Y is aggregate output, K is the aggregate capital stock, L is aggregate labor
supply and A is a technology parameter. Thesubscript t denotes the time period.
The aggregate capital stock depends on aggregate investment I and the depreci-
ation rate δ:
Kt+1 = (1−δ)Kt +It with 0 ≤ δ ≤ 1 (2)
The productivity parameter A follows a stochastic path with trend growth g and
an AR(1) stochastic component:
∗ ˆ
lnA = lnA +A
t t t
Aˆt = φAAˆt−1 +εA,t with |φA| < 1 (3)
∗ ∗
A = A (1+g)
t t−1
The stochastic shock ε is i.i.d. with mean zero.
A,t
The goods market always clears, such that the firm always sells its 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
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following first-order-conditions:
Y
(1−α) t = w (4)
t
L
t (cid:2) (cid:3) (cid:2) (cid:3)
1 Yt+1 1−δ
1 = E α +E (5)
t t
1+rt+1 Kt+1 1+rt+1
According to equation (4), the firm hires labor until the marginal product of
labor is equal to its marginal cost (which is the real wage w). Equation (5) shows
that the firm’s investment demand at time t is such that the marginal cost of
investment, 1, is equal to the expected discounted marginal product of capital at
time t+1 plus the expected discounted value of the extra capital stock which is
left after depreciation at time t+1.
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The Stochastic Growth Model The stochastic growth model
The government The government consumes every period t an amount G ,
t
which follows a stochastic path with trend growth g and an AR(1) stochastic
component:
∗ ˆ
lnG = lnG +G
t t t
Gˆt = φGGˆt−1 +εG,t with |φG| < 1 (6)
∗ ∗
G = G (1+g)
t t−1
The stochastic shock ε is i.i.d. with mean zero. ε and ε are uncorrelated
G,t A G
at all leads and lags. The government finances its consumption by issuing public
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debt, subject to a transversality condition, and by raising lump-sum taxes. The
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timing of taxation is irrelevant because of Ricardian Equivalence.
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The Stochastic Growth Model The stochastic growth model
The representative household There is one representative household, who
derives utility from her current and future consumption:
(cid:4) (cid:8)
(cid:6) (cid:7)
(cid:5)∞ 1 s−t
U = E lnC with ρ > 0 (7)
t t s
1+ρ
s=t
The parameter ρ is called the subjective discount rate.
Every period s, the household starts off with her assets X and receives interest
s
payments X r . She also supplies L units of labor to the representative firm, and
s s
therefore receives labor income w L. Tax payments are lump-sum and amount to
s
T . She then decides how much she consumes, and how much assets she will hold
s
in her portfolio until period s+1. This leads to her dynamic budget constraint:
Xs+1 = Xs(1+rs)+wsL−Ts −Cs (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:
(cid:4)(cid:9) (cid:11) (cid:8)
(cid:10)s
1
lim Et Xs+1 = 0 (9)
s→∞ s(cid:2)=t 1+rs(cid:2)
This equation is called the transversality condition.
The household then takes X and the current and expected values of r, w, and T
t
as given, and chooses her consumption path to maximize her utility (7) subject
to her dynamic budget constraint (8) and the transversality condition (9). This
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leads to the following Euler equation:
(cid:2) (cid:3)
1 1+rs+1 1
= E (10)
s
Cs 1+ρ Cs+1
Equilibrium Every period, the factor markets and thegoods market clear. For
the labor market, we already implicitly assumed this by using the same notation
(L) for the representative household’s labor supply and the representative firm’s
labor demand. Equilibrium in the goods market requires that
Y = C +I +G (11)
t t t t
Equilibrium in the capital market follows then from Walras’ law.
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The Stochastic Growth Model The steady state
3. The steady state
Let us now derive the model’s balanced growth path (or steady state); variables
∗
evaluated on the balanced growth path are denoted by a .
ˆ
To derive the balanced growth path, we assume that by sheer luck ε = A =
A,t t
ε = Gˆ = 0, ∀t. The model then becomes a standard neoclassical growth
G,t t
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model, for which the solution is given by:
(cid:6) (cid:7)
α
∗ α 1−α ∗
Y = A L (12)
t r∗ +δ t
(cid:6) (cid:7)
1
∗ α 1−α ∗
K = A L (13)
t r∗ +δ t
(cid:6) (cid:7)
1
∗ α 1−α ∗
I = (g +δ) A L (14)
t r∗ +δ t
(cid:2) (cid:3)(cid:6) (cid:7)
α
C∗ = 1−(g +δ) α α 1−α A∗L−G∗ (15)
t r∗ +δ r∗ +δ t t
(cid:6) (cid:7)
α
w∗ = (1−α) α 1−α A∗ (16)
t r∗ +δ t
r∗ = (1+ρ)(1+g)−1 (17)
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The Stochastic Growth Model Linearization around the balanced growth path
4. Linearization around the balanced growth path
Let us now linearize the model presented in section 2 around the balanced growth
path derived in section 3. Loglinear deviations from the balanced growth path
are denoted by aˆ(so that Xˆ = lnX −lnX∗).
Below are 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
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equation (10) and the equilibrium condition (11):
Yˆ = αKˆ +(1−α)Aˆ (18)
t t t
1−δ g +δ
ˆ ˆ ˆ
Kt+1 = Kt + It (19)
1+g 1+g
ˆ
Y = wˆ (20)
(cid:2) (cid:3)t t
Et rt1+1+−r∗r∗ = r1∗++rδ∗ (cid:12)Et(Yˆt+1)−Et(Kˆt+1)(cid:13) (21)
(cid:2) (cid:3)
Cˆt = Et(cid:12)Cˆt+1(cid:13)−Et rt1+1+−r∗r∗ (22)
∗ ∗ ∗
C I G
Yˆ = t Cˆ + t Iˆ + t Gˆ (23)
t ∗ t ∗ t ∗ t
Y Y Y
t t t
The loglinearized laws of motion of A and G are given by equations (3) and (6):
ˆ ˆ
At+1 = φAAt +εA,t+1 (24)
ˆ ˆ
Gt+1 = φGGt +εG,t+1 (25)
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The Stochastic Growth Model Solution of the linearized model
5. Solution of the linearized model
I now solve the linearized model, which is described by equations (18) until (25).
ˆ ˆ ˆ ˆ
First note that K , A and G are known in the beginningof periodt: K depends
t t t t
ˆ ˆ
on past investment decisions, and A and G are determined by current and past
t t
ˆ ˆ ˆ
values of respectively ε and ε (which are exogenous). K , A and G are
A G t t t
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
ˆ ˆ
maximize their profits and utility (I and C are therefore called period t’s control
t t
variables); the values of the interest rate and the wage are such that they clear
the capital and the labor market.
Solving the model requires that we express period t’s endogenous variables as
ˆ
functions of period t’s state variables. The solution of C , for instance, therefore
t
looks as follows:
ˆ ˆ ˆ ˆ
C = ϕ K +ϕ A +ϕ G (26)
t CK t CA t CG t
The challenge now is to determine the ϕ-coefficients.
First substitute equation (26) in the Euler equation (22):
ˆ ˆ ˆ
ϕ K +ϕ A +ϕ G
CK t CA t CG t (cid:2) (cid:3)
= Et(cid:12)ϕCKKˆt+1 +ϕCAAˆt+1 +ϕCGGˆt+1(cid:13)−Et rt1+1+−r∗r∗ (27)
Now eliminate Et[(rt+1−r∗)/(1+r∗)] with equation (21), and use equations (18),
ˆ ˆ ˆ
(24) and (25) to eliminate Yt+1, At+1 and Gt+1 in the resulting expression. This
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The Stochastic Growth Model Solution of the linearized model
ˆ
leads to a relation between period t’s state variables, the ϕ-coefficients and Kt+1:
ˆ ˆ ˆ
ϕ K +ϕ A +ϕ G
CK(cid:6)t CA t CG t (cid:7) (cid:6) (cid:7)
∗ ∗
r +δ r +δ
= ϕCK +(1−α)1+r∗ Kˆt+1 + ϕCA −(1−α)1+r∗ φAAˆt +ϕCGφGGˆt
(28)
Wenowderiveasecondrelationbetweenperiodt’sstatevariables,theϕ-coefficients
ˆ ˆ
and Kt+1: rewrite the law of motion (19) by eliminating It with equation (23);
ˆ ˆ
eliminate Y and C in the resulting equation with the production function (18)
t t
and expression (26); note that I∗ = K∗(g +δ); and note that (1−δ)/(1 +g)+
∗ ∗ ∗
(αY )/(K (1+g)) = (1+r )/(1+g). This yields:
t t
(cid:2) (cid:3)
∗ ∗
1+r C
Kˆt+1 = 1+g − K∗(1+g)ϕCK Kˆt
(cid:2) (cid:3) (cid:2) (cid:3)
(1−α)Y∗ C∗ G∗ C∗
+ − ϕ Aˆ − + ϕ Gˆ
K∗(1+g) K∗(1+g) CA t K∗(1+g) K∗(1+g) CG t
(29)
ˆ
Substituting equation (29) in equation (28) to eliminate Kt+1 yields:
ˆ ˆ ˆ
ϕ K +ϕ A +ϕ G
CK(cid:2) t CA t CG t(cid:3)(cid:2) (cid:3)
∗ ∗ ∗
r +δ 1+r C
= ϕ +(1−α) − ϕ Kˆ
CK 1+r∗ 1+g K∗(1+g) CK t
(cid:2) (cid:3)(cid:2) (cid:3)
r∗ +δ (1−α)Y∗ C∗
+ ϕ +(1−α) − ϕ Aˆ
CK 1+r∗ K∗(1+g) K∗(1+g) CA t
(cid:2) (cid:3)(cid:2) (cid:3)
∗ ∗ ∗
r +δ G C
− ϕ +(1−α) + ϕ Gˆ
CK 1+r∗ K∗(1+g) K∗(1+g) CG t
(cid:6) (cid:7)
∗
r +δ
+ ϕ −(1−α) φ Aˆ −ϕ φ Gˆ (30)
CA 1+r∗ A t CG G t
ˆ ˆ ˆ
As this equation must hold for all values of K , A and G , we find the following
t t t
system of three equations and three unknowns:
(cid:2) (cid:3)(cid:2) (cid:3)
∗ ∗ ∗
r +δ 1+r C
ϕ = ϕ +(1−α) − ϕ (31)
CK CK 1+r∗ 1+g K∗(1+g) CK
(cid:2) (cid:3)(cid:2) (cid:3)
r∗ +δ (1−α)Y∗ C∗
ϕ = ϕ +(1−α) − ϕ
CA CK 1+r∗ K∗(1+g) K∗(1+g) CA
(cid:6) (cid:7)
∗
r +δ
+ ϕ −(1−α) φ (32)
CA 1+r∗ A
(cid:2) (cid:3)(cid:2) (cid:3)
∗ ∗ ∗
r +δ G C
ϕ = − ϕ +(1−α) + ϕ −ϕ φ
CG CK 1+r∗ K∗(1+g) K∗(1+g) CG CG G
(33)
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