advmacro Takeki Sunakawa HP

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Hayashi and Prescott (2002):

“The 1990s in Japan: A Lost Decade”

Takeki Sunakawa

Advanced Macroeconomics at Tohoku University

T. Sunakawa Hayashi and Prescott (2002) Advanced Macro 1 / 26

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Introduction

The Japanese economy has been stuck in depression since the 1990s. The average growth of GDP per capita is 0.5% for 1991-2000.

What is the problem?

Inadequate fiscal policy, over investment in the 1980s, problems with financial intermediation, etc.

A breakdown of financial system in 1997-98 is not the cause of the problem.

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Introduction, cont’d

The problem is a low growth rate of total factor productivity (TFP).

The reduction of the workweek length also contributes to the low growth rates.

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Poor performance in the 1990s

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Workweek falls in the 1998-93 period

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Capital deepens...

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...as the rate of return declines

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Government expenditure and investment

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Production function

The representative firm’s production function is given by: Yt= AtK_t^θ(htEt)^1−θ,

where Ytis aggregate output, Atis the TFP, Ktis aggregate capital, htis hours per worker (workweek), and Etis employment.

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Growth accounting

Let Ntbe the working-age population and define y = Y /N , e = E/N , and x = K/Y . Then we have

Yt= AtK_t^θ(htEt)^1−θ,

⇔ Y_t^1/(1−θ)= A^1/(1−θ)_t K_t^θ/(1−θ)htEt,

⇔ Yt= A^1/(1−θ)_t x^θ/(1−θ)_t htEt,

∴ _y_t_{= A}^1/(1−θ)_t _x^θ/(1−θ)_t _h_t_e_t_.

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Growth accounting, cont’d

Growth of output per capita is decomposed as

gyt= ¹

1 − θ^g^At⁺ θ

1 − θ^g^xt^{+ g}^ht^{+ g}^et^, where θ = 0.362.

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Household’s utility

There are Nthouseholds in working age. The utility function is X∞

t=0

Nt(log ct− g(ht; zt)et) ,

where et= Et/Ntis the fraction of working households and ct= Ct/Nt is consumption per capita.

Labor is indivisible so that a person eigher works ht hours or does not work at all.

Households also choose htif they work. ztis the number of days worked and exogenously given.

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Household’s budget constraint

The household’s budget constraint is given by

Ct+ It ≤ wthtEt+ rtKt− τ (rt− δ)Kt− πt, It = Kt+1− (1 − δ)Kt,

where Ctis aggregate consumption, Itis aggregate investment, wtis wage rate and rt is real interest rate.

A distortionary tax is imposed at rate τ on capital income after depreciation. πt is a lumpsum tax.

Second welfare theorem does not hold.

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Closing the model

Resource constraint is given by

Ct+ It+ Gt= Yt.

The government’s budget constraint is implicitly obtained as Gt= τ (rt− δ)Kt+ πt.

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Solving the model: Firm

The representative firm maximizes its profit: (marginal cost) = (marginal product)

wt= (1 − θ)yt/(htet), rt= θyt/kt.

where yt and ktare output and capital per capita.

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Solving the model: Household

The household’s Lagrangean is

Lt≡ X∞ t=0

β^t{Nt(log ct− g(ht; zt)et)} .

+λtNt(wthtet+ rtkt− τ (rt− δ)kt− ˜πt− ct− ntkt+1+ (1 − δ)kt) , where nt≡ Nt+1/Nt and ˜πt≡ πt/Nt.

The FOCs are

∂ct: c⁻_t¹= λt,

∂et: g(ht; zt) = λtwtht,

∂ht: g^′(ht; zt) = λtwt,

∂kt+1: 1 = β^λ^t+1 λt

(1 + (1 − τ )(rt+1− δ)) .

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Solving the model: Aggregation

The resource constraint (per capita) is

ct+ ntkt+1− (1 − δ)kt+ gt= yt. The production function is

yt= Atk^θ_t(htet)^1−θ.

We have {ct, kt, yt, ht, et, λt, wt, rt} and 8 equations, so we can solve the model.

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Equilibrium conditions

The equlibrium conditions are summerized as ctg(ht; zt) = (1 − θ)yt/et, ctg^′(ht; zt) = (1 − θ)yt/(htet), 1 = β ^c^t

ct+1

(1 + (1 − τ )(rt+1− δ)) , ct+ ntkt+1− (1 − δ)kt+ gt= yt, yt= Atk^θ_t(htet)^1−θ,

wt= (1 − θ)yt/(htet), rt= θyt/kt.

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Labor disutility and hours worked

Combining the first two equations

hg^′(h; z) = g(h; z),

where g(h; z) is convex in h and g(0; z) > 0 because of the fixed cost of going work. h(z) is determined separately from the rest of the model. Further, g(h) is approximated around ¯h = 40 as

g(h(z)) ≈ ^αh(z)_¯ h ^. [Note what HP actually do is somewhat different.]

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Indivisible labor

Why g(h) is convex and g(0) > 0? etis also the probability of employed, and the household’s utility function is

log(ct) + φ(ht)et+ φ0(1 − et),

where φ^′(h) < 0 and φ^′′(h) < 0 (e.g., φ(h) = A log(1 − h)). Ignoring a constant term, g(h) ≡ φ0− φ(h).

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How hours worked are determined

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Detrending

Let Zt≡ A_t^1−θ¹ and define

˜

ct≡ ct/Zt, ^˜kt≡ kt/Zt, y˜t≡ yt/Zt, γt≡ Zt+1/Zt, ψt≡ gt/yt, nt≡ Nt+1/Nt. Finally, we have

(αht/¯h)˜ct= (1 − θ)˜yt/(htet), 1 = ^β

γt

˜ ct

˜

ct+1(1 + (1 − τ )(rt+1− δ)) ,

˜

ct+ γtnt^˜kt+1− (1 − δ)˜kt= (1 − ψt)˜yt,

˜

yt= ˜k^θ_t(htet)^1−θ, rt= θ˜yt/˜kt.

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Calibration

The model is calibrated using the data during 1984-89 (which is found in columns FX-GB in FED_data.xls).

θ = 0.362: average of the capital income share in GNP.

δ = 0.089: ratio of depreciation to the beginning-of-the-year capital stock. τ = 0.480: average of the capital tax.

β = 0.976: obtained from the Euler equation ct+1

ct = β (1 + (1 − τ )(θyt+1/kt+1− δ)) . α = 1.373: obtained from the wage equation so that h(zt) = ¯h (?)

αct= (1 − θ)yt/et= wtht.

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Simulation

The exogenous variables are (At, Nt, ψt) where ψt= Gt/Yt. For the 1990s (t = 1990, 1991, ..., 2000), actual values are used. For t = 2001, 2002, ...,

The growth rate of A^1/(1−θ)_t is set to its 1991-2000 average of 0.29%. Ntis 2000 value; no population growth.

ψtis set to 1999-2000 value of 15%.

Deterministic simulation is done. The decline of TFP growth in the 1990s is forecasted in 1990.

Also, in our Dynare simulation, we set h(zt) = ¯h = 40 for all t.

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Figure 6-8 in the paper

1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 80

100

120 Detrended y

1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 1.5

2 2.5

Capital-output ratio

1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 0.02

0.04 0.06

0.08 After-tax rate of return

actual HP Dynare

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Some comments

Detrended output is close to the model predictions.

Capital-output ratio rises and the rate of return falls as output growth falls, which is seen in the Euler equation.

The model’s prediction is sensitive to the exogenous variables in the 1990s. The most important variable is TFP.

Dynare simulation yields almost the same result as in HP.