Appendix to the Solow model
Takeki Sunakawa
Advanced Macroeconomics at Tohoku University
What is covered
Stochastic simulations (McCandless sec. 1.4) Log-linearization (McCandless sec. 1.5) Detrending the Solow residual
Stochastic TFP
We assume that the TFP follows a stochastic process: log At+1= (1 − ρ) log ¯A + ρ log At+ εt+1, where εt∼ N (0, σε2).
Note that
At+1= ¯A1−ρAρteεt+1, holds.
The stochastic process of At is estimated by OLS using a time series of TFP. A typical quarterly estimate for the U.S. economy is ρ = 0.95 and σε= 0.008.
Fundamental equation of economic growth
The fundamental equation of economic growth is
(1 + n)kt+1= (1 − δ)kt+ σAtf (kt),
where f (k) = kθ. [How to derive this equation?] The steady state is given by
k =¯
σ ¯A n + δ
1−θ1 .
Log-linearization
We approximate the model around the steady state. Use the formula of approximation
xt≡ x exp ˆxt≈ ¯x(1 + ˆxt),
where ¯x is the steady state of xtand ˆxt is percent deviation from the steady state.
Log-linearization: Production function
Production function:
yt≡ Atf (kt) = Atkθt. It can be written as
¯
y exp(ˆyt) = ¯A¯kθexp(ˆat+ αˆkt). In the steady state, ¯y = ¯A¯kθholds. Then,
ˆ
yt= ˆat+ θˆkt. [Note: This is not approximation.]
Log-linearization: Resource constraint
Resource constraint:
(1 + n)kt+1= (1 − δ)kt+ σyt. It can be written as
(1 + n)¯k exp(ˆkt+1) = (1 − δ)¯k exp(ˆkt) + σ ¯y exp(ˆyt). Use the formula of approximation
(1 + n)¯k(1 + ˆkt+1) = (1 − δ)¯k(1 + ˆkt) + σ ¯y(1 + ˆyt).
In the steady state, (1 + n)¯k = (1 − δ)¯k + σ¯y holds. Then we have (1 + n)¯kˆkt+1= (1 − δ)¯kˆkt+ σ ¯y ˆyt.
Log-linearization: Summary
After all, the log-linealized equlibrium conditions are: ˆ
yt= ˆat+ θˆkt,
(1 + n)¯kˆkt+1= (1 − δ)¯kˆkt+ σ ¯y ˆyt. Or,
(1 + n)¯kˆkt+1= (1 − δ)¯kˆkt+ σ ¯y(ˆat+ θˆkt).
First-order difference equation
It can be rewritten as the first-order difference equation: ˆkt+1= Bˆkt+ Cˆat, where
B = 1 − δ + σθ(¯y/¯k)
1 + n ,
C = σ(¯y/¯k) 1 + n .
Analytical solution for the variance
Assume ρ = 0. Recursively substituting, we have kˆt+1= C
∞
X
i=0
Biεt−i.
With this expression, the variance of capital around the steady state is given by
varˆk = C2σ2ε 1 − B2.
Simulations
1.9 2 2.1 2.2 2.3 2.4 2.5 2.6
exact log-linear
Solow residual
Identifying the aggregate technology shock with the Solow residual: log Zt= log Yt− α log Kt− (1 − α) log Nt. log Zthas a trend. How to remove the trend?
Solow residual
-9.6 -9.5 -9.4 -9.3 -9.2 -9.1 -9 -8.9
annual log tfp
Data source (NIPA and CPS)
GDP: Table 2A. Real Gross Domestic Product > Gross domestic product (Line 1)
Capital: Table 5.9. Changes in Net Stock of Produced Assets (Fixed Assets and Inventories) > Private (Line 2)
GDP deflator: Table 1.4.4. Price Indexes for Gross Domestic Product, Gross Domestic Purchases, and Final Sales to Domestic Purchasers > Gross domestic product (Line 1)
Hours worked: Cociuba, Prescott and Uberfeldt “U.S. Hours and Productivity Behavior Using CPS Hours Worked Data: 1947-III to 2011-IV”
Linear trend
Remove linear trend: zt= log Zt− a − bt where a and b are obtained by OLS.
Hodrick-Prescott filter
Let yt be a time series and
yt= gt+ ct, where gtis trend and ct is cyclical component.
The Hodrick-Prescott filter solves the following problem:
{gmint}Tt=1
( T X
t=1
(yt− gt)2+ λ
T−1
X
t=2
[(gt+1− gt) − (gt− gt−1)]2 )
,
where λ is smoothing parameter.
Hodrick-Prescott filter, cont’d
FOCs are
∂g1: c1= λ(g3− 2g2+ g1),
∂g2: c2= λ(g4− 2g3+ g2) − 2λ(g3− 2g2+ g1),
∂gt: ct= λ(gt+2− 2gt+1+ gt) − 2λ(gt+1− 2gt+ gt−1) +λ(gt− 2gt−1+ gt−2)
for t = 3, 4, ..., T − 2,
∂gT−1 : cT−1= −2λ(gT− 2gT−1+ gT−2) + λ(gT−1− 2gT−2+ gT−3),
∂gT : cT = λ(gT − 2gT−1+ gT−2).
Hodrick-Prescott filter, cont’d
FOCs are
∂g1: c1= λ(g3− 2g2+ g1),
∂g2: c2= λ(g4− 4g3+ 5g2− 2g1),
∂gt: ct= λ(gt+2− 4gt+1+ 6gt− 4gt−1+ gt−2) for t = 3, 4, ..., T − 2,
∂gT−1: cT−1= λ(−2gT + 5gT−1− 4gT−2+ gT−3),
∂gT : cT = λ(gT − 2gT−1+ gT−2).
Matrix form
In a matrix form, c = y − g = λFg where
F=
1 −2 1 0 · · · 0
−2 5 −4 1 0 · · · 0
1 −4 6 −4 1 0 · · · 0
0 1 −4 6 −4 1 0 · · · 0
... ...
... ...
0 · · · 0 1 −4 6 −4 1 0
0 · · · 0 1 −4 6 −4 1
0 · · · 0 1 −4 5 −2
0 · · · 0 1 −2 1
.
Then, g = (I − λF)−1y.
Cyclical component
0.94 0.96 0.98 1 1.02 1.04 1.06 1.08
Linear HP
Trend
-9.7 -9.6 -9.5 -9.4 -9.3 -9.2 -9.1 -9 -8.9
annual log tfp
Assignment #1
Let n = .02, δ = .1, θ = .36 and σ = .2. Also let ¯A = 1, ρ = 0 and σε= .2.
1 Simulate the model for 1,000 periods and compute var(k).
2 Compare it with the analytical solution for the variance.
3 Do 1-2 with 100,000 period simulation.
4 What about the case of ρ >0? Try to derive the analytical solution for the variance.