Master in EconomicsLecture 1: Small Open Economies

International Business Cycle

Jose Ignacio LopezHEC Paris

October 2015ENSAE

Content of the class

The class will consist of 3 separate sections:

1 International Business Cycles (Lopez)

2 Exchange Rates and International Portfolios (Michalski)

3 Sovereign Debt (Mengus)

International Business Cycles

For the �rst part of the class we will cover 4 main topics

1 Small Open Economies

2 Two-country models

3 International risk sharing

4 Models with heterogeneous agents

Key Questions

• What are the features and causes of business cycles in emergingeconomies (are they di�erent from developed economies?)

• Can standard business cycles models explain internationaltransmission of shocks and cycles in regions/set of countries?

• Do countries share risk? How e�ectively?

• What are the propagation mechanisms of international shocks?Do micro-structure matters?

Small Open Economies• Small Open Economies (SOE) are countries for which

international trade/ �nance are an important part of theeconomy and international prices are taken as given (terms oftrade, interest rates, oil prices) .

• SOE also is used to describe models that are not general, butpartial equilibrium as one variable (e.g. interest rates) isexogenous to the system

• When comparing stylized features of SOE business cycles thereis a pattern that emerges between high and middle/low incomecountries (e.g. Canada vs Argentina).

• The labels of “developed” and “emerging” drawn fromper-capita income levels are very often used (S&P and IFCclassi�cation) but have been recently criticized (e.g. Portugal vsChile or South Korea vs Spain) [link FT Article]

Small Open Economies Business Cycles

Emerging Economies Developed Marketsσ (Y ) 2.74 1.34σ (4Y ) 1.87 .95ρ (Y ) .76 .75σ (C ) /σ (Y ) 1.45 .94σ (I ) /σ (Y ) 3.91 3.41ρ (TB/Y ,Y ) -51 -.17ρ (C ,Y ) .72 .66

Notes: Averages for 13 emerging and 13 developed economies. Quarterly data detrended usingHodrick-Prescott �lter. The emerging economies are: Argentina, Brazil, Ecuador, Israel, Ko-rea, Malaysia, Mexico, Peru, Philippines, Slovak Republic, South Africa, Thailand, Turkey. De-veloped countries: Australia, Austria, Belgium, Canada, Denmark, Finland, Netherlands, NewZeleand, Norway, Portugal, Spain, Sweden, Switzerland. Source: Aguiar and Gopinath (2007)

Standard SOE

• The starting point of the standard SOE is an endowmenteconomy (see [Notes Schmitt-Grohe Uribe])

• In�tinitely-lived representative household that maximizesconsumption (c) subject to stochastic exogenous endowment(y) and with access to international capital markets in the formof non-contigent (risk-free) real debt (d) that pays a constantinterest rate (r).

max{ct ,dt}∞t=0E0

∞∑t=0

βtU (ct) (1)

dt = (1 + r) dt−1 + ct − yt (2)

limj→∞dt+j

(1 + r)j ≤ 0 (3)

Consumption Dynamics

U (ct) = β (1 + r) EtU (ct+1) (4)

• We need to assume β (1 + r) = 1 otherwise consumption willdrift permanently (long-rung growth or decay)

• Let’s assume quadratic preferences:

U (ct) = −12

(ct − c)2 (5)

• The Euler equation becomes:

ct = Etct+1 (6)

Intertemporal Resource Constraint• Iterating forward the budget constraint at time t , one can �nd

the intertemporal resource constraint of the economy

(1 + r) dt−1 =dt+j

(1 + r)j +

j∑s=0

yt+s − ct+s

(1 + r)s (7)

• Applying conditional expectations time t , taking the limitj →∞ and using the transversality condition

(1 + r) dt−1 = Et

∞∑s=0

yt+s − ct+s

(1 + r)s (8)

• The country’s initial net foreign debt position equals theexpected discount of current and future savings.

rdt−1 + ct =r

1 + rEt

∞∑s=0

yt+s

(1 + r)s (9)

Equilibrium Allocation

• Following the literature, let’s assume the endowment is anAR(1) process:

yt = ρyt−1 + εt (10)

• With ρ < 1, εt i.d.d. innovations. We solve for consumption:

ct =r

1 + r − ρyt − rdt−1 (11)

• The Trade Balance (tbt = yt − ct)

tbt =1− ρ

1 + r − ρyt + rdt−1 (12)

CA and Debt Dynamics

• Current Account (cat ≡ − (dt − dt−1) = −rdt−1 + tbt )

cat =1− ρ

1 + r − ρyt (13)

• The evolution of the debt is

dt = dt−1 −1− ρ

1 + r − ρyt (14)

Main features SOE

• If shocks are not perfectly persistent (ρ < 1) :

• Consumption responds less than output• Trade Balance and CA improve upon positive shock

(pro-cyclical)• Debt falls with a positive shock

• If shocks are perfectly persistent (ρ→ 1) :

• Consumption increases one-to-one• Trade Balance and CA remain unchanged• Debt remains constant

• The endogenous variables of the model are Random Walks.

• There are 3 ways to induce stationarity often used in theliterature (see Schmitt-Grohé and Uribe (2003))

Solving the Stationarity Problem

• Endogenous Discount Factor: The discount factor (β) is notconstant but a function of consumption: β (ct) with βc < 0. Insteady-state β (c) (1 + r) = 1, which pins down thesteady-state level of consumption as function of r and theparameters de�ning β (c)

• e.g: U (c) = c1−γ

1−γ and β (c) = [1 + c]−ψ1 as in Mendoza (1991)

• Debt Elastic Interest Rate Premium: the interest rate isincreasing in the net foreign debt position: β (1 + rt (dt)) = 1.This helps to select the level of debt in steady-state.

• e.g: rt = r + p (dt) with p (dt) = ψ2

(edt−d − 1

).

Solving the Stationarity Problem II

• Convex portfolio adjustment costs:

dt = (1 + r) dt−1 + ct − yt +ψ3

2(dt − d

)2.

• The Euler equation becomes:

U (ct)[1− ψ3

(dt − d

)]= β (1 + r) U (ct+1)

• In steady-state:β (1 + r) = 1

dt = d

Adding Capital to the model

• In the standard SOE the only way to smooth shocks over time isby adjusting the trade balance.

• In reality, households have access to domestic technologies tosave.

Non-stochastic Model with Capital

max{ct ,it ,bt}∞t=0

∑βtU (ct) (15)

bt = (1 + r) bt−1 + yt − ct − it (16)

yt = AtF (kt) (17)

kt+1 = kt + it (18)

limj→∞bt+j

(1 + r)j ≥ 0 (19)

FOC

λt = β (1 + r)λt+1 (20)

λt = βλt+1[At+1F ′k (kt+1) + 1

](21)

bt = (1 + r) bt−1 + AtF (kt)− ct − kt+1 + kt (22)

With the assumption β (1 + r) = 1

ct = ct+1 (23)

The two equilibrium conditions:

r = At+1F ′k (kt+1) (24)

ct = rbt−1 +r

1 + r

∞∑s=0

At+sF ′k (kt+s)− kt+s+1 − ks

(1 + r)s (25)

Steady-state

Because there is no depreciation, investment in steady-state is zero.All the variable in the model are determined by A and r.

css = c iss = 0 yss = AF(k)

bss = c−yr kss = f

(A, r)

cass = 0

tbss = −r b

(26)

Permanent Productivity Shocks

• Consider A′ > A in period 0 and thereafter.• Consumption increases permanently. Capital stock should

increase but it is �xed at time 0. Investment jumps so theeconomy reaches the new k ′.

• The trade balance deteriorates at time 0 and improves at time 1.

c0 = r b +r

1 + r[A′F

(k)

+(k ′ − k

)]+

11 + r

A′F(K ′)

tb0 = −r b − 11 + r

[A′F

(k ′)− A′F

(k)

+(k ′ − k

)]

Transitory Productivity Shocks• Consider A′ > A in period 0 and thereafter goes back to A .• Consumption increases temporarily but the capital stock

remains unchanged (and so investment). The extra-output isused to invest in international markets . The trade balanceimproves at time 0 and deteriorates thereafter.

c0 = c−1 +r

1 + r[A′ − A

]F(k)

(27)

tb0 − tb−1 =1

1 + r[(

A′ − A)F(k)]

(28)

• The more persistent the productivity shocks, the more likelythe trade balance will experience a deterioration at the time of apositive shock.

• Adjustment costs of capital could o�set this initial e�ect becauseinvestment will increase slowly

• How much persistence and how large adjustment costs? This isa quantitative question

Real Business Cycle Model (Mendoza1991)

• Let’s consider the standard RBC model as in Mendoza (1991)

max{ct ,it ,bt}∞t=0E0

∞∑t=0

βtU (ct) (29)

dt = (1 + rt) dt−1 − yt + ct + it + Φ (kt+1 − kt) (30)

ln (At+1) = ρln (At) + εt+1 (31)

with the following functional forms:

U (c , h) =

[c− hω

ω

]1−γ−1

1−γ rt = r + ψ2

(edt−d − 1

)y = Akαh1−α Φ (kt+1 − kt) = φ (kt+1−kt)2

2

Calibration RBC

Parameter Value Description

γ 2 Inter-temporal elasticity of substitutionω 1.45 Labor Supply Elasticityψ2 .000742 Discount Factor Parameterα .32 Capital Shareφ 0.028 Adjustment Costβ 0.96 Real Interest Rateδ .1 Depreciation Rateρ .42 Persistence TFP shockσε 0.0129 Volatility Innovations TFPd .7442 Debt

Notes: Source: Mendoza (1991)

Impulse Response Functions

5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025GDP

5 10 15 20 25 30 35 400

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018Consumption

5 10 15 20 25 30 35 40-0.02

0

0.02

0.04

0.06

0.08

0.1Investment

5 10 15 20 25 30 35 400

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018Hours

5 10 15 20 25 30 35 40

×10-3

-10

-8

-6

-4

-2

0

2

4

6

8Trade Balance/GDP

5 10 15 20 25 30 35 40

×10-3

-8

-6

-4

-2

0

2

4

6

8Current Account/GDP

Small Open Economies Business Cycles

Canada ModelVolatility σx/σy ρx,y Volatility σx/σy ρx,y

y 2.8 1 1 3.08 1 1c 2.5 0.89 0.59 2.71 0.88 0.84i 9.8 3.50 0.64 9.04 2.94 0.66h 2 0.71 0.8 2.12 0.69 1tby 1.9 0.68 -0.13 1.78 0.58 -0.04

Notes: Own calculations and Mendoza (1991)

Role of Persistence and Adjustment Costs

Figure : Impulse Response Function TB/Y (productivity shock)

The Cycle is the Trend (Aguiar-Gopinath2007)

• Shocks to the trend of productivity could be the primary sourceof �uctuations

• This requires to distinguish between transitory and permanentshocks.

Model Aguiar-Gopinath 2007• Period Utility Household (Cobb-Douglas consumption and

leisure):

Ut =

[C γ

t (1− Lt)1−γ]1−σ

1− σ• Cobb-Douglas production function with temporary and

permanent productivity shocks:Yt = eztK 1−α

t (ΓtLt)α

Γt =t∏

s=0

egt

zt = ρzzt−1 + εzt gt = (1− ρg )µg + ρggt−1 + εgt

• Budget constraint (with quadratic adjustment cost ofinvestment)

Ct+Kt+1 = Yt+(1− δ) Kt−φ

2

(Kt+1

Kt− eµg

)2

Kt−Bt+qtBt+1

Debt Elastic Interest Rate

• Interest Rate depends on the level of debt (target the level ofnormalized debt b)

1qt

= 1 + rt = 1 + r∗ + ψ

[exp

(Bt+1

Γt− b)− 1]

• Variable need to be de-trended to render them stationary:xt = xt/Γt−1

• The de-trended (and recursive) problem:

V = max{C ,L,K ′,B′}

[C γ

t (1− Lt)1−γ]

1− σ

1−σ

+ βegγ(1−σ)EV ′

C + eg K ′ = Y + (1− δ) K − φ2

(eg K ′

K− eµg

)2K − B + egqB ′

FOC (without hats) and Steady-Stateγ U

c = λ

(1− γ) U1−L = αY

L

λeg(

1 + φ

(eg K ′

K− eµ

))= βegγ(1−σ)λ′[(1− α)

Y ′

K ′+ (1− δ)]

− φ

2

(egK ′′

K ′− eµg

)2

+φegK ′′

K ′

(egK ′′

K ′− eµg

)

λegq = βegγ(1−σ)λ′[1 + r + ψ

(eB−b − 1

)]• Steady-state:

1 = βeµg [γ(1−σ)−1][(1− α) Y

K + (1− δ)]

1 = βeµg [γ(1−σ)−1] (1 + r)

C + K (eµg + δ − 1) = Y + B(egq − 1)

Calibration Aguiar and Gopinath

Parameter Value Description

β .98 Time Preferencesγ .36 Labor equals one-thirdb 0.1 Normalized debtψ 0.001 Interest Rate Premiumα 0.68 Labor Shareσ 2 Risk Aversionδ .5 Depreciation Rateφ 4 Capital Adjustment Costµg log(1.0066) Calibrated for Mexicoρz 0.95 Calibrated for Mexicoρg 0.01 Calibrated for Mexicoσεg 0.0281 Calibrated for Mexicoσεz 0.048 Calibrated for Mexico

Notes: Source: Aguiar and Gopinath (2007)

Emerging Economy Business Cycles

Mexico Model

σy 2.40 2.39σ4(y) 1.52 1.72σc/σy 1.26 1.27σi/σy 4.15 2.59σtb/y/σy 0.90 0.71ρy 0.83 0.78ρy,tb/y -0.75 -0.66ρc,y 0.92 0.95ρc,y 0.91 0.92

Notes: Own calculations and Aguiar and Gopinath (2007)

Impulse-Response Functions

5 10 15 20 25 30 35 40-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Trade Balance /GDP

5 10 15 20 25 30 35 40-0.8

-0.6

-0.4

-0.2

0

0.2

0.4Consumption/GDP

5 10 15 20 25 30 35 40-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2Investment/GDP

5 10 15 20 25 30 35 40-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2Trade Balance/GDP

5 10 15 20 25 30 35 40-0.2

-0.1

0

0.1

0.2

0.3Consumption /GDP

5 10 15 20 25 30 35 40-0.2

0

0.2

0.4

0.6

0.8

1

1.2Investment / GDP

Transitory (z) Permanent (g)

Figure : IRF (permanent and transitorily productivity shock)

The Role of Interest Rate Shocks(Perri-Neumeyer 2005)

• Credit conditions change easily for emerging econonies

• Interest rates are correlated to business cycles

.

Model Perri-Neumeyer 2005• Period Utility Household (Greenwood, Hercowitz, and Hu�man

(1988) preferences over consumption and leisure):

Ut =

[ct − ψ (1 + γ)t lνt

]1−σ1− σ

• Cobb-Douglas production function with with working capital:

yt = Atkαt((1 + γ)t lt

)1−απt = yt − wt lt − rtkt − [Rt − 1] θwt lt

At = ρAAt−1 + εAt Rt = R∗D

• Budget constraint (with quadratic adjustment cost ofinvestment)

ct + it + bt+1 + κ (bt+1) = wt lt + rkt kt + btRt

Perri-Neumeyer 2005

• Interest Rate depends on a premium for risk assets R∗ (commonto all countries) and a country-speci�c default D

Rt = R∗t Dt R∗ = ρ1R∗t−1 + εR∗

t Dt = ρ1Dt−1 + εDt

• Investment and Portfolio adjustment costs:

it = kt − (1− δ) kt−1 +φ

2kt−1

(kt

kt−1− (1 + γ)

)2

κ (bt+1) =κ

2yt+1

(bt+1

yt+1− b)2

FOC Perri-Neumeyer 2005

(ct − ψlνt )−σ = λ

ψνlν−1t = wt = (1− α) yt

lt

(1

1+rtθ

)

λ

[1 + φ

((1 + γ) k ′

k− (1 + γ)

)]= β (1 + γ)−σ λ′[(1− α)

y ′

k ′+ (1− δ)

− φ

2((1 + γ) k ′′/k ′ − (1 + γ)

)2+φk ′′

k ′

((1 + γ) k ′′

k ′− (1 + γ)µg

)]

λ[1 + κ

(b′y ′ − b

)]= β (1 + γ)−σ λ′R ′

Interest Rate Shock

Figure : Labor Markets with GHH and Cobb-Douglas Preferences

Calibration Perri-Neumeyer

Parameter Value Description

β 0.93 Time Preferencesσ 5 Risk Aversionν 1.6 Labor Disutilityψ 2.48 Labor Weightγ 0.0062 Growth Rateα 0.38 Capital Shareδ 0.044 Depreciation Rateθ 1 Labor Paid in Advanceκ 10−5 Bond Cost Holdingφ 25.1 Adjustment cost Capital

At = ρAAt−1 + εAt ρA = 0.95, σεA = 0.95 Productivity ShocksR∗t = ρ1R∗t−1 + εRt ρ1 = 0.81, σεR = 0.63% International RateDt = ρ2Dt−1 + εDt ρ2 = 0.78, σεD = 2.59% Country Risk

Source: Neumeyer and Perri (2005)

Impulse-Response Functions

Figure : IRF (Interest Rate Shock)

ReferencesAguiar, M. and G. Gopinath (2007). Emerging market business

cycles: The cycle is the trend. Journal of Political Economy 115(1).Greenwood, J., Z. Hercowitz, and G. W. Hu�man (1988). Investment,

capacity utilization, and the real business cycle. The AmericanEconomic Review, 402–417.

Mendoza, E. G. (1991). Real business cycles in a small open economy.The American Economic Review, 797–818.

Neumeyer, P. A. and F. Perri (2005). Business cycles in emergingeconomies: the role of interest rates. Journal of monetaryEconomics 52(2), 345–380.

Schmitt-Grohé, S. and M. Uribe (2003). Closing small open economymodels. Journal of international Economics 61(1), 163–185.