advmacro Takeki Sunakawa IRBC 1

International Real Business Cycles: Part I

Takeki Sunakawa

Advanced Macroeconomics at Tohoku University

International RBC

Goods and assets are traded across national borders.

How much of these connections have impact on aggregate fluctuations? The dynamic general equilibrium theory has contributed so far to explain the data. However, the theory remains significantly different from the data. There are two anomalies in termsquantitiesandprices.

Two anomalies

Quantity anomaly: Cross country correlations of output are larger than those of consumption and productivity.

Price anomaly: The terms of trade is very persistent and highly volatile. The model cannot generates the volatility observed in the data.

Business cycles, 1970-mid-1990

Within countries

Volatility: consumption and output has about the same standard deviation. Investment is about two to three times more volatile than output.

Employment is less volatile than output.

Output volatility ranges from 0.90 (France) to 1.92 (Switzerland and U.S.) Consumption volatility (relative to that of output) ranges from 0.66 (Australia) to 1.15 (U.K.). Note the measure includes consumer durables. Persistence: Output is highly persistence, from 0.57 (Austria) to 0.90 (Switzerland).

Correlation: Employment is positively correlated with output, orprocyclical, in all countries. Other variables are also procyclical, except government purchases and the trade balance (the ratio of net exports to output).

International comovements, 1970-mid-1990

Across countries

Output: 0.40-0.70 for Japan and European countries. 0.76 for Canada. Consumption: The correlations are smaller than those of output. [Government purchases: Mostly positive but small correlations.]

Models

We will consider two models.

In one model, two countries produce a single homogeneous good (Backus, Kehoe and Kydland, 1992).

Time-to-build investment structure (Kydland and Prescott, 1982) is also introduced.

In the other, each country produce different, imperfectly substitutable goods, which accounts for the relative price of two goods (Backus, Kehoe and Kydland, 1994).

Households

The preferences of the consumer in country i = 1, 2 are given by

u_i= E0

∞

X

t=0

β^tU (c_it, 1 − n_it),

where citand nitare consumption and employment in country i. The functional form of the preferences is

U (c, 1 − n) =c^µ(1 − n)^{1−µ1−γ}/(1 − γ). Note that when γ = 1, U (c, 1 − n) = µ log c + (1 − µ) log(1 − n).

Production and resources

Output in country i is given by

yit= zitF (kit, nit) = zitk_it^θn^1−θ_it The world resource constraint is

X

i

(cit+ xit+ git) =^X

i

yit,

where xitis investment and gitis the government purchases for country i.

Stochastic shocks

The vectors zt= (z1_t, z2_t) and gt= (g1_t, g2_t) are stochastic shocks to productivity and government purchases.

The productivity shock follows

zt+1= Azt+ ε^z_t+1, where ε^z_t = (ε^z1_t, ε^z2_t) ∼ N (0, Vz).

Similarly, shocks to the government purchase follows gt+1= Bgt+ ε^g_t+1, where ε^g_t = (ε^g1_t, ε^g2_t) ∼ N (0, Vg).

Planner’s problem

The second theorem holds; the competitive equilibrium is Pareto optimal. The social planner maximizes u1+ u2 subject to the technology and the resource constraint.

The same weight is assigned on each country’s preference. We set up the Lagrangean

L = E0

∞

X

t=0

β^t{U1_t+ U2_t

+λt

X

i

(zitF (kit, nit) + (1 − δ)kit− cit− kit+1− git) )

.

FONCs

The FONCs are

∂cit: ^∂Uit

∂cit

= λt,

∂nit: −^∂Uit

∂nit

= zit

∂Fit

∂nit

λt,

∂kit+1: λit= βEt

 λt+1

 zit+1

∂Fit+1

∂kit+1

+ (1 − δ)

 , for i = 1, 2.

There are 7 variables, {cit, nit, kit+1, λt}, and 7 equations.

Equilibrium conditions

Using the functional forms U (c, 1 − n) =c^µ(1 − n)^{1−µ1−γ}/(1 − γ) and F (k, n) = k^θn^1−θ,

λt=c^µ_it(1 − nit)^{1−µ1−γ µ} cit

, 1 − µ

µ cit

1 − nit

= (1 − θ)^yit nit

, 1 = βEt

 λ_it+1 λit

 θ^yit+1

kit+1

+ (1 − δ)

 , yit= zitk^θ_itn¹⁻_it ^θ,

kit+1= (1 − δ)kit+ xit, for i = 1, 2 and

Xyit=^X(cit+ xit+ git) .

Steady state

In steady state, the two countries are symmetric: 1 − µ

µ c

1 − n ^{= (1 − θ)} y

n^{, 1 = β}

θ^y

k^{+ 1 − δ}

, y = k^θn^1−θ, δk = x,

y = c + x + g, Then we have

y k ⁼

β⁻¹− 1 + δ

θ ^,

k n ⁼

_y k

θ₋₁¹

, y

n ⁼

 k n

θ

, ^c

y ^{= 1 − δ} k y ⁻

g y^, Ψ := ^{1 − n}

n ⁼

1 − µ (1 − θ)µ

c

y^, n = (1 + Ψ)⁻¹,

Log-linearization

Log-linearized equations are

λˆt+ γˆcit+ (1 − γ)(1 − µ)(ˆcit+ ⁿ

1 − n^{nˆit) = 0,} ˆ

cit+ ⁿ

1 − n^{nˆit= ˆyit− ˆnit,}

λˆt− Etλ^ˆt+1= (1 − β(1 − δ))^Etyˆit+1− ˆkit+1

, ˆ

yit= ˆzit+ θˆkit+ (1 − θ)ˆnit, kˆit+1= (1 − δ)ˆkit+ δ ˆxit, X

i

ˆ yit=^X

i

 c y^ˆcit+

δk y ^xˆit+

g y^ˆgit

 .

Net exports

The volume of net exports is given by

nxit= yit− cit− xit− git.

in steady state, nx = 0. Cannot take logs of negative values!

Time-to-build investment structure

Investment for J periods is required in advance to install capital in period t + J:

kit+1 = (1 − δ)kit+ s¹_it, s^j_it+1 = s^j+1_it

for j = 1, ..., J − 1. Investment at date t is

xit=

J

X

j=1

φjs^j_it=

J

X

j=1

φjs¹_it+j−1,

where we set φ_j = 1/J.

Note that s^J_it−1= s^{J −1}_it = ... = s¹_{it+J −2} are predetermined in period t. This

Planner’s problem: Time-to-build

We set up the Lagrangean

L = E0

∞

X

t=0

β^t{U1_t+ U2_t

+λt

X

i



zitF (kit, nit) − cit−

J

X

j=1

φjs¹_it+j−1− git





+^X

i

µit (1 − δ)kit+ s¹_it− kit+1

)

.

What are the control variables?

FONCs: Time-to-build

The FONCs are

∂cit: ^∂Uit

∂cit

= λt,

∂nit: −^∂Uit

∂nit

= zit

∂Fit

∂nit

λt,

∂kit+J : µit+J −1= βEt

 zit+J

∂Fit+J

∂k_it+J^{λt+J+ (1 − δ)µit+J}

 ,

∂s¹_{it+J −1}: µit+J −1=

J

X

j=1

β^−j+1φjλt+J⁻j

for i = 1, 2.

There are 11 variables, {cit, nit, kit+J, sit+J −1, µit+J −1, λt+J}, and 11

Equilibrium conditions: Time-to-build

Using the functional forms,

λt=c^µ_it(1 − nit)^{1−µ1−γ µ} cit

, 1 − µ

µ cit

1 − nit

= (1 − θ)^yit nit

, 1 = βEt

µit+J

µit+J −1

 θ^yit+J

kit+J

λt+J

µit+J

+ (1 − δ)

 ,

µ_{it+J −1}=

J

X

j=1

β^−j+1φ_jλ_{t+J −j}, yit= zitk_it^θn¹⁻_it ^θ,

kit+1= (1 − δ)kit+ s¹_it, xit=

J

Xφjs¹_it+j−1,

Parameter values

Taken from Table 3 in the paper (We also set J = 1 for our benchmark case):

Volatility and persistence

Volatility

Economy Output Net Exp. Consp. Invest. Emp. Prod.

US 1.92% .52% .75 3.27 .61 .68

Euro 1.01 .50 .83 2.09 .85 .98

Bench, J = 1 2.19 29.14 .32 31.39 .54 .49 Bench, J = 4 1.54 4.32 .44 10.05 .49 .69

20 simulations of 100 periods are done with Dynare.

Correlation

Correlation with output Economy Consp. Invest. Net Exp. Emp. Prod.

US .82 .94 -.37 .88 .96

Euro .81 .89 -.25 .32 .85

Bench, J = 1 .77 .01 .05 .97 .84

Bench, J = 4 .80 .29 .00 .93 .90

Correlation between countries Economy Output Consp. Invest. Emp. Prod.

US vs. Euro .66 .51 .53 .33 .56

Bench, J = 1 -.53 .72 -1.00 -.88 .32 Bench, J = 4 -.15 .90 -.96 -.76 .32

The effects of a productivity shock in Home

0 0.5 1 1.5 2

z₁ y₁ c₁ x₁ n₁

The effects of a productivity shock in Foreign

-1 -0.5 0 0.5 1

z₂ y₂ c₂

Some notes

Trading frictions (“Transport Cost”) can reduce volatility in investment and net exports. Also, they can make investment (but not net exports) procyclical (see Table 5 in the paper).

X

i

(cit+ xit+ git) =^X

i

yit−^X

i

τ (nxit)²,

where τ = .1/y and y is the steady state value of output (BKK, 1992 p. 769).

However, correlations across countries are not much affected. Especially, consumption correlation is very high, whereas output correlation is very low, even negative. This is also calledthe correlation puzzle.

If γ = 1, preferences areadditively separable, which leads to a perfect correlation of consumption across countries.

Exercise

Replicate the results with time-to-build structure with J = 4.