Master in EconomicsLecture 2: Two-Country Models

International Business Cycle

Jose Ignacio LopezHEC Paris

October 2015ENSAE

Two-Country Models

• Two-country models ask whether extensions ofclosed-economy models that explain domestic business cyclescan also help to understand the international dimension ofaggregate �uctuations

• The cannonical paper is Backus, Kehoe, and Kydland (1992) thatextends Kydland and Prescott (1982) to a two-country setup

• The key question is whether a single aggregate productivityshock can explain at the same time the beviour of domestic andinternational variables

Two-Country Model (BKK 1992)

• Two countries, each represented by a large number ofin�tinitely-lived representative household that maximizesconsumption (c) and leisure (1− l) subject to stochasticproductivity shocks at home (Z ) and abroad (Z ∗).

• The production function is Cobb-Douglas

max{ct ,lt}∞t=0E0

∞∑t=0

βt

[cµt (1− lt)

1−µ]1−σ

1− σ(1)

yt = Ztkαt l1−αt (2)

nxt = yt − ct − kt+1 + (1− δ) kt (3)

The Planner’s Problem• The planner’s problem is the maximization of the weigthed sum

(Pareto-Negishi Ψ ) of each country’s utility subject theproduction functions and the aggregate resource constraint:

Max{ct ,c∗t ,lt ,l∗t ,kt+1,k∗t+1} [ΨU (ct , lt) + (1− Ψ)U (c∗t , l∗t )] (4)

yt + y∗t = ct + c∗t + it + i∗t (5)• The planner equalizes marginal utility of consumption:

ΨUc (c , l) = (1− Ψ)Uc (c∗, l∗)

• The other FOC are standard:(1− µ)µ

c1− l

= (1− α) yl

λ = βλ′(α

y ′

k ′+ 1− δ

)

Decentralized Problem (Home)• This problem can be decentralized by assuming there is a full

set of state-contigent claims (Arrow-Debrew securities insequential markets)

• An event st ∈ S is a realization of productivities (potentiallyin�nite set). A history is a collection of events: st = (s0, .., st).

• The optimization problem of the domestic household is

Max∞∑

t=0

∑st

βtπ(st) [c (st)µ (1− l (st))1−µ

]1−σ1− σ

(6)

w(st) l

(st)+ r

(st) k

(st)+ b

(st) = c

(st)+ k

(st+1)

+ (1− δ) k(st)

+∑st+1

q(st , st+1

)b(st , st+1

)

Descentralized Problem (Foreign)

• The foreign household maximizes

Max∞∑

t=0

∑st

βtπ(st) [c (st)∗µ (1− l∗ (st))1−µ

]1−σ1− σ

(7)

• Subject to the following budget constraint:

w∗(st) l∗

(st)

+r∗(st) k∗

(st)+ b∗

(st) = c∗

(st)+ k∗

(st+1)

+ (1− δ) k∗(st)

+∑st+1

q(st , st+1

)b∗(st , st+1

)

Optimal Conditions

λ(st) = Uc

(st) (8)

q(st , st+1

)= β

π (st , st+1)

π (st)

Uc (st , st+1)

Uc (st)(9)

1 =∑st+1

βπ (st , st+1)

π (st)

λ (st , st+1)

λ (st)

[r(st)+ 1− δ

](10)

1− µµ

c (st)

1− l (st)= w

(st) (11)

Uc (st , st+1)

Uc (st)=

U∗c (st , st+1)

U∗c (st)(12)

Calibration BKK Model

Parameter Value Description

β 0.99 Time Preferencesµ 0.34 Consumption Shareσ 2 Intertemporal Elasticityα 0.36 Capital Shareδ 0.5 Depreciation Rate[

ZtZ ∗t

]= A

[Zt−1Z ∗t−1

]+

[εtε∗t

]

A =

[a11 a12a21 a2

] [0.906 0.0880.088 0.906

]Productivity Shocks

ρε,ε∗ 0.258 Correlation Shocksσ2ε = σ2

ε∗ 0.00852 Variance Shocks

Notes: Backus, Kehoe, and Kydland (1992)

Impulse Response Functions

0 10 20 30 40 50-2

-1

0

1

2

3

4

5

6

7Home

ConsumptionOutputInvestmentProductivityHours

0 10 20 30 40 50-5

-4

-3

-2

-1

0

1

2

3

4Foreign

Figure : IRS (percentage deviations of steady-state)

Business Cycles (Domestic Variables)

σx/σGDP ρx,GDP σx/σGDP ρx,GDP σx/σGDP ρx,GDP

GDP 1 1 1 1 1 1C 0.75 0.82 0.83 0.81 0.42 0.77I 3.27 0.94 2.09 0.89 10.99 0.27L 0.61 0.88 0.85 0.32 0.50 0.93Z 0.68 0.96 0.98 0.85 0.67 0.89nx 0.27 -0.37 0.49 -0.25 2.51 0.01

Notes: Backus, Kehoe, and Kydland (1995)

Business Cycles (InternationalCo-movement)

US and Europe ModelInternational Correlations

GDP 0.66 -0.21C 0.51 0.88I 0.53 -0.94L 0.33 -0.78Z 0.56 0.25

Notes: Backus, Kehoe, and Kydland (1995)

BKK (1995)• Each country specializes in the production of intermediate

goods (a for home, b for foreign):

a1t + a2t = yt = Ztkαt l1−αt

b1t + b2t = y∗t = Z ∗t k∗αt l∗1−αt

• The two goods are aggregated using an Armington Aggregatorwith elasticity of substitution 1/θ

ct + it =[ωa1−θ

1,t + (1− ω) b1−θ1,t

] 11−θ

• If pa1t and pb

1t are the prices of domestic and foreign goods inunits of domestic �nal good, the terms of tradeare:tott =

pa1t

pb1t

=(

a1tb1t

)θ1ω

• The trade balance to GDP ratio and the RER are:

nxt =pa1ta2t−pb

1tbtyt

rert =pa1t

pa2t

=pb1t

pb2t

Calibration Heathcote and Perri (2002)

Parameter Value Description

β 0.99 Time Preferencesµ 0.34 Consumption Shareσ 2 Intertemporal Elasticityα 0.36 Capital Shareδ 0.5 Depreciation Rateis 0.15 Import Share1/θ 0.9 Elasticity of Substitutionρε,ε∗ 0.290 Correlation Shocksσε = σε∗ 0.0073 Variance Shocks

A =

[a11 a12a21 a2

] [0.970 0.0250.025 0.970

]Productivity Shocks

Notes: Heathcote and Perri (2002)

Business Cycles (two-goods BKK model)

Data Modelσx/σGDP ρx,GDP ρx ,x∗ σx/σGDP ρx,GDP ρx ,x∗

GDP 1 1 0.58 1 1 0.18C 0.81 0.86 0.36 0.53 0.96 0.65I 2.84 0.95 0.30 2.74 0.96 0.29L 0.66 0.87 0.42 0.31 0.97 0.14nx 0.27 -0.49 0.43 -0.64tot 1.79 -0.24 0.61 0.65rer 4.02 0.13 0.45 0.65

Notes: Heathcote and Perri (2002)

The Role of the Elasticity of Substitution

Figure : E�ects of Varying the Elasticity of Substitution on the Volatility ofTOT and Import Ratio

Notes: Backus et al. (1995)

The Role of the Financial Structure

• Heathcote and Perri (2002) study the e�ects of departing fromthe complete market assumptions: bond market economy and�nancial autarky

• The only element that changes in the set up of the problem isthe budget constraint of households

• Bond Economy:

pa1(st) [w (st) l

(st)+ r

(st) k

(st)] = c

(st)+ i

(st)

+pa1(st) b

(st)

+ pa1(st) q

(st) b

(st+1)

• Financial Autarky:

pa1(st) [w (st) l

(st)+ r

(st) k

(st)] = c

(st)+ i

(st)

Equilibrium Conditions under di�erent�nancial structures

• Complete Markets

q(st) = β

π (st , st+1)

π (st)

Uc (st , st+1)

Uc (st)

pa1 (s

t , st+1)

pa1 (st)

rer(st , st+1

)= k

U∗c (st , st+1)

Uc (st , st+1)

where κ = rer (s0)Uc (s0) /U∗c (s0)• Bond Economy

q(st) = 1

1 + r (st)= β

∑st+1

π (st , st+1)

π (st)

Uc (st , st+1)

Uc (st)

pa1 (s

t , st+1)

pa1 (st)

b(st) = b∗

(st)

• Financial Autarky: nxt = 0

Business Cycle Statistics (Domestic)

Business Cycle Statistics (Cross-Country)

Impulse Response Functions

ReferencesBackus, D., P. Kehoe, and F. Kydland (1992). International real

business cycles. Journal of Political Economy 100(4), 745.Backus, D. K., P. J. Kehoe, and F. Kydland (1995). International

business cycles: Theory vs. evidence," in thomas f. cooley, ed.,frontiers of business cycle research, princeton university press.

Heathcote, J. and F. Perri (2002). Financial autarky and internationalbusiness cycles. Journal of Monetary Economics 49(3), 601–627.

Kydland, F. and E. Prescott (1982). Time to build and aggregate�uctuations. Econometrica 50(6), 1345–1370.