Master in EconomicsLecture 3: International Risk Sharing

International Business Cycle

Jose Ignacio LopezHEC Paris

October 2015ENSAE

Risk Sharing andInternational Business Cycles

• Inspecting the data alone cannot answer what is the degree ofrisk sharing (e�ciency) in international markets

• The Two-Country model can be used to assess whether thecross-country allocation of resources is e�cient (Pareto sense)

• One key aspect in this discussion is the cross-countrycorrelation of consumption.

Cole-Obstfeld (1991)

• Ilustration of perfect risk sharing in a world without completemarkets

• Two-country endowment economy in which countryspecializes in the production of one commodity

• Identical preferences over the two-commodities.

CH =(cH1

)α (cH2

)1−α

• Demand functions:

cH1 =αYp1

cH2 =(1−α )Y

p2cF1 =

αY ∗p1

cF2 =(1−α )Y ∗

p2

• Resource constraint: cH1 + cF1 = Y cH2 + c

F2 = Y ∗

• Prices: p1 =α (Y+Y ∗)

Y p2 =(1−α ) (Y+Y ∗)

Y ∗

• Allocation: CH

CH+F =p1Y

p1Y+p2Y ∗= α CH

CF =α

1−α

International Risk Sharing (Quantities)

• Output, investment, and employment co-move positively andstrongly across countries.

• The cross-country correlation of consumption is positive butsmaller than the correlation of output.

• Net exports are not very volatile (their standard deviation isabout one-third that of GDP) and are strongly countercyclical.

E�ciency - Endowment Economy• Discussion of e�ciency in the BKK model: Heathcote and Perri

(2014)

• Example Endowment economy with two-goods.

Maxκc

1−γ1

1 − γ+ (1 − κ)

c1−γ2

1 − γ

(1)

subject to:a1 + a2 = z1

b1 + b2 = z2

c1 =[ωa

σ−1σ

1 + (1 − ω) bσ−1σ

1

] σσ−1

c2 =[(1 − ω) a

σ−1σ

2 + ωbσ−1σ

2

] σσ−1

E�cient Allocation• FOC

ωκc−γ+ 1

σ1 a

− 1σ

1 = (1 − ω) (1 − κ) c−γ+ 1

σ2 a

− 1σ

2

(1 − ω) κc−γ+ 1

σ1 b

− 1σ

1 = ω (1 − κ) c−γ+ 1

σ2 b

− 1σ

2

• Log-linear system:

sa1 + (1 − s ) a2 = z1

(1 − s ) b1 + (s ) b2 = z2(−γ + 1

σ

)c1 −

1σ a1 =

(−γ + 1

σ

)c2 −

1σ a2(

−γ + 1σ

)c1 −

1σ b1 =

(−γ + 1

σ

)c1 −

1σ b2

E�cient Allocation (Solution)

Solution

c1 − c2y1 − y2

=(2s − 1)

2γ (s − 1) + 4s (σγ − 1) (1 − s ) + 1

The model �ts the datac1 − c2y1 − y2

> 1

ifσ < σ =

1γ−

(1 − γ2sγ

)Note that when s = ω, γ = 1, σ = 1 (Cole-Obstfeld)

Risk Aversion andthe Elasticity of Substitution

BKK with intermediate goods

Max

∑i=1,2

(cµi(1 − li )1−µ

)1−γ

1 − γ

(2)

subject to:a1 + a2 = Z1F (k1, l1)

b1 + b2 = Z2F (k2, l2)

c1 + k′1 − (1 − δ ) k1 = G1 =

[ωa

σ−1σ

1 + (1 − ω) bσ−1σ

1

] σσ−1

c2 + k′2 − (1 − δ ) k2 = G2 =

[ωa

σ−1σ

2 + (1 − ω) bσ−1σ

2

] σσ−1

[Z1tZ2t

]=

[ρ ψψ ρ

] [Z1t−1Z2t−1

]+

[ε1t−1ε2t−1

]

E�cient Conditions

Uc1Ga1 = Uc2Ga2

Uc1Gb1 = Uc2Gb2

λ1 = βλ′1 [Ga1Fk1 + (1 − δ )]

λ2 = βλ′2 [Gb2Fk2 + (1 − δ )]

Ul1 = Uc1Ga1Fl1

Ul2 = Uc2Gb2Fl2

E�ciency in the BKK (revisited)

International Risk Sharing (Prices)

• The correlation with the consumption ratio(cH

cF

)and the real

exchange rate (Q = eP∗

P ) for various pairs of developedcountries is close to zero, or even negative (Backus-SmithPuzzle)

• In the model the correlation is close to one.

Uc2

Uc1

=Ga1

Ga2= Q

• With separable preferences:(c1c2

)γ=

Ga2

Ga1= Q

The Backus-Smith Puzzle

The Backus-Smith Puzzle (II)

• Mussa (1986) documents that the real exchange rate is muchmore volatile between countries with �exible exchange rates

• Hess and Shin (2010) and Devereux and Hnatkovska (2011)report that the correlation between exchange rates and relativeconsumption is negative for countries (regions) with �exibleexchange rates and positive for countries sharing a �xedexchange rate

• The connection between nominal exchange rates andfundamentals remains elusive

Asset Pricing View Exchange Rates

• Using the standard model:

Uc2t

Uc2t= Qt

Uc2t+1

Uc2t+1= Qt+1

• Brandt et al. (2006) propose the following decomposition:

qt+1 − qt = log

(Uc2t+1

Uc2t

)− log

(Uc1t+1

Uc1t

)= log (m2t ) − log (m1t )

• Colacito and Croce (2011) use Epstein-Zin preferences todistangle the SDF with movements in consumption.

ReferencesBrandt, M. W., J. H. Cochrane, and P. Santa-Clara (2006).

International risk sharing is better than you think, or exchangerates are too smooth. Journal of Monetary Economics 53(4),671–698.

Colacito, R. and M. M. Croce (2011). Risks for the long-run and thereal exchange rate. Journal of Political Economy 119(1).

Devereux, M. B. and V. Hnatkovska (2011). Consumptionrisk-sharing and the real exchange rate: Why does the nominalexchange rate make such a di�erence?

Heathcote, J. and F. Perri (2014). Handbook of internationalEconomics, Volume 4. Elsevier.

Hess, G. D. and K. Shin (2010). Understanding the backus–smithpuzzle: It’s the (nominal) exchange rate, stupid. Journal ofInternational Money and Finance 29(1), 169–180.

Mussa, M. (1986). Nominal exchange rate regimes and the behaviorof real exchange rates: Evidence and implications. InCarnegie-Rochester Conference series on public policy, Volume 25,pp. 117–214. Elsevier.