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Appendix to the Solow model

Takeki Sunakawa

Advanced Macroeconomics at Tohoku University

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What is covered

Stochastic simulations (McCandless sec. 1.4) Log-linearization (McCandless sec. 1.5) Detrending the Solow residual

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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.

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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 .

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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.

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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.]

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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.

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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).

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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 .

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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.

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Simulations

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6

exact log-linear

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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?

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Solow residual

-9.6 -9.5 -9.4 -9.3 -9.2 -9.1 -9 -8.9

annual log tfp

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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”

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Linear trend

Remove linear trend: zt= log Zt− a − bt where a and b are obtained by OLS.

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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.

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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).

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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).

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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.

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Cyclical component

0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

Linear HP

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Trend

-9.7 -9.6 -9.5 -9.4 -9.3 -9.2 -9.1 -9 -8.9

annual log tfp

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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.

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