Risk Sharing among Large and Small Countries · IntroductionModelEdgeworth BoxWelfare and...

Introduction Model Edgeworth Box Welfare and Reversals Two-Factor Model Conclusions

Risk Sharing among Large and Small Countries

Giancarlo CorsettiUniversity of Cambridge and CEPR

Anna LipińskaFederal Reserve Board

December 12, 2017

The views expressed here should not be interpreted as reflecting the views ofthe Federal Reserve System.


Question and Motivation

We study perfect risk sharing among asymmetric countries.

Why?

• Countries differ in business cycle and structuralcharacteristics.

• Policy relevance:• Debate on policies to enhance risk sharing;

(Fatas 1998; Farhi and Werning 2017; Beblavy et al 2015)

• Measurement of degree of international risk sharing;(Asdrubali et al 1996)

• Literature did not focus on implications of asymmetries forrisk sharing.


Our Approach and Main Result

Our framework: a two-period model of 2 countries that:

• have identical initial endowment,• differ in size and stochastic properties of output.

Perfect risk sharing among asymmetric countries results in anefficient redistribution of consumption:

• Asymmetries imply that the shadow price of domesticstate-contingent output differs across borders.

• An average level of consumption under perfect risk sharingmay fall (or rise) relative to its level under financial autarky.


Relevance

• Derivation of perfect risk sharing allocation in large modelswith asymmetric countries is hard!

• Many studies impose symmetry of equilibrium which impliesequalization of consumption levels (”income pooling”).

• Differences in the price of risk due to asymmetries amongcountries are then forced not to matter in determiningper-capita consumption.

• This practice may lead to spurious “welfare reversals”:welfare is higher under financial autarky than under completemarkets.


Further ImplicationsEmpirics and Political Economy

• Consumption growth rates may not be equalized at the timeof a risk-sharing enhancing reform.

• Empirical studies that measure degree of risk sharing byinvestigating correlation of consumption growth rates amongcountries should take this into account.

• Efficient redistribution due to risk sharing relevant to politicalequilibria on institutional and market reforms.(Persson and Tabellini 1996).

• Pursuing risk sharing via income pooling implements transfersto offset the income effects of adjustment in the equilibriumprice of risk.


Related Literature

• Early theoretical discussion of the issue in quantitative studies:

(Cole and Obstfeld 1991; Chari et al 2002; Devereux and Engel 2003; Monacelli

and Faia 2004; Tille and Pesenti 2004)

• Spurious welfare reversals and approximation errors:

(Kim and Kim 2003).

• Optimal Portfolio choice literature:

(Devereux and Sutherland 2007).


Outline

• Model (one factor).

• Complete Markets, financial autarky, and perfect pooling.

• Edgeworth box analysis with country size.

• Welfare and reversals.

• Generalization (two factors).

• Spurious Welfare Reversals in a DSGE model.


Analytical FrameworkTwo-Period Endowment Model

• First example: asymmetric volatility—a single global shockwith different factor loadings for each country.

• In second example we show that results generalize tocountry-specific shocks.

• We contrast three different arrangements:

1. complete markets (CM),2. financial autarky (FA),3. income pooling (IP).

• Welfare analysis under the second order approximation,U(Ci ) = log(Ci ) where i = H,F .


Model Specification

• Size: H has population n; F has (1-n).• Endowment is deterministic in period 1:

YH,1 = YF ,1 = 1.

• Endowment is stochastic in period 2:

YH,2 = 1− γH�,YF ,2 = 1− γF �.

where E (�) = 0, E (�2) = σ2� and γH , γF are factor loadings.

• Aggregate output:YW ,1 = 1.

YW ,2 = nYH,2 + (1− n)YF ,2 = 1− γW �.

where γW = nγH + (1− n)γF .


Complete Markets AllocationNonlinear Solution

• Consumption growth rates in each country are equalisedacross all states of nature.

• In levels, consumption in each country is a fraction of theworld endowment (Obstfeld and Rogoff 1996):

Ci ,t = µiYW ,t .

where µi =

Yi,1YW ,1

+βE

(Yi,2YW ,2

)1+β and i = H,F .


Complete Markets AllocationFirst and Second Order Approximation

Ci ,t = µiYW ,t .

where µi =

Yi,1YW ,1

+βE

(Yi,2YW ,2

)1+β and i = H,F .

• Up to first order approx:E(

YH,2YW ,2

)≈ 1 implying that CH,t = CF ,t , in both periods

t = 1, 2.

• Up to second order approx:E(

YH,2YW ,2

)≈ 1 + (γW − γH)γWσ2� .

µH = 1 +βγW (γW − γH)

1 + βσ2� , µF = 1 +

βγW (γW − γF )1 + β

σ2� .


Efficient Consumption ShareAnalytical Representation

µi = 1 +βγW (γW − γi )

1 + βσ2�

where i = H,F and γW = nγH + (1− n)γF .

• Up to second order, µi can be higher or lower than 1:

• µi = 1 if γW = 0 or γW = γH .

• Home and Foreign consumption share increase in the difference(γW − γi ).

• γi can be negative which can result in negative γW .

• γW is a function of the size of the country: the larger thecountry the larger its influence on the world output and γW .


Efficient Consumption Share

-2 -1 0 1 2

.H while .

F=1

0.5

1

1.5

2

2.5n0=0.01

-2 -1 0 1 2

.H while .

F=1

0.4

0.6

0.8

1

1.2

1.4n0=0.25

-2 -1 0 1 2

.H while .

F=1

0.6

0.8

1

1.2

1.4n0=0.5

-2 -1 0 1 2

.H while .

F=1

0.5

1

1.5

2

2.5n0=0.75

-2 -1 0 1 2

.H while .

F=1

0

1

2

3

4n0=0.99

7H

7F

n7H

+(1-n)7F


Different Risk-Sharing ArrangementsLevels vs Growth Rates of Consumption

• Consumption LEVELS are equalized under IP:

CH,t = CF ,t where t = 1, 2.

...as well as (by construction) under FA in period 1:

CH,1 = 1 = CF ,1(recall: CH,2 = 1− γH� 6= CF ,2 yet CH = CF = 1.)

• may not be equalized under CM, because of the endogenousadjustment of income to the price of risk:

CH,1 = µH ,CH,2 = µHYH,2 with CH = µH .

• Consumption GROWTH RATES instead are equalizedunder both CM and IP, not under FA.


Edgeworth Box Analysis

Unconventional use of the box:

• Individual preferences and endowment of representative agentsof two countries H and F.

• When we allow the number of agents in H to be infinitesimal:World price of risk dictated by autarky price in F.

• In the graph to follow, we set γH > γF and γF = 1.Recall that Yi,2 subject to a mean zero shock.


Edgeworth BoxFinancial Autarky

E

H

FPreference and endowment of representative agents in country H and F

output in state 2

output in state 1

output in state 1

output in state 2


Edgeworth BoxBudget Constraint at World Prices that coincide with F Autarky Prices

E

then FFA prices=world prices.

H

F

FAH

world prices

Size of H ->0; size of F->1


Edgeworth BoxComplete Markets when H and F differ in size

E

CMH

Clearly, at the complete market prices, Home gains: CMH>FAH

H

F

FAH

Agents in the large country keep consuming their endowment


Edgeworth BoxConsumption under Income Pooling (H and F differing in size)

E=IPF

CMH

H

F

FAH

Now, suppose H and F ''pool income"

From the vantage point of H, IPH mirrors IPF All agents consume the same average income in F: E=IPF

IPH


Complete Markets vs Income PoolingHome has high volatility of output, with γH = 2

Income Pooling is a good deal for Home

E=IPF

IPH

CMH

IPH>CMH>FAH

H

F

FAH

The price of risk is equalised worldwide at FFA prices


Income Pooling vs Complete MarketsIP may however be a bad deal for Home

Home output variance high but negative covariance with F γH = −2, (γF = 1)

γH=-2 implies CMH>IPH>FAH

IPH

CMH

E=IPF

H

F


Financial Autarky vs Income PoolingHome may even be better off in FA

volatility of output not too high, e.g. γH = 0.5, (γF = 1)

IPH

ECMH

γH=.5 implies CMH>FAH>IPH

H

F


Welfare Analysis in the General CaseSecond Order Approximation

We now reconsider welfare ranking using:

E (log(CH)) = E (log(CH)) +1

CHE (CH − CH)−

1

C2H

E (CH − CH)2

Home loss under different risk sharing arrangements:

• FA: LFAH =β2γ

2Hσ

2� .

• CM: LCMH = −β(γW − γH)γWσ2� +β2γ

2Wσ

2� .

• IP: LIPH =β2γ

2Wσ

2� .


Ranking Welfareas a Function of Size and Volatility

Table 1: Ranking of CM, FA, IP for Home and Foreign countrydepending on home size n and factor loading γH (setting γF = 1):

n γH0.01 < −199 (−199,−99) (−99,−0.98) (−0.98, 1) > 10.25 < −7 (−7,−3) (−3,−0.6) (−0.6, 1) > 10.5 < −3 (−3,−1) (−1.− 0.33) (−0.33, 1) > 1

0.75 < −1.67 (−1.67,−0.33) (−0.33,−0.14) (−0.14, 1) > 10.99 < −1.02 (−1.02,−0.01) (−0.01,−0.005) (−0.005, 1) > 1home IP � CM � FA IP � CM � FA CM � IP � FA CM � FA � IP IP � CM � FA

foreign CM � FA � IP CM � IP � FA IP � CM � FA IP � CM � FA CM � IP � FA

• For −99 < γH < −.01 (given γF = 1), implying negativecovariance of output, the Home country prefers:• complete markets if its size is small (n=0.01).• income pooling if its size is large (n=.99).


Generalization: Two Factor ModelModel Description

• Deterministic endowment in period 1:

YH,1 = YF ,1 = 1.

• Stochastic endowment in period 2:

YH,2 = 1− �H ,YF ,2 = 1− �F .

YW ,2 = nYH,2 + (1− n)YF ,2 = 1− n�H − (1− n)�F .

• E (�H) = E (�F ) = 0, E (�2H) = σ2�H

, E (�2F ) = σ2�F

,E (�H�F ) = σ�H�F .

• U(Ci ) = log(Ci ) where i = H,F .


Complete Market Allocationin the Two-Factor Model

britishsss

• Consumption in each country is a fraction of the world

endowment with µi =

Yi,1YW ,1

+βE

(Yi,2YW ,2

)1+β .

• E(

YH,2YW ,2

)≈ 1 + n(n− 1)σ2�H + (1− n)

2σ2�F + 2n(1− n)σ�H�F .

• Home consumption share:

µH = 1 +β(1− n)(n(σ�H�F − σ2�H )− (1− n)(σ�H�F − σ

2�F

))

1 + β.

• if n→ 0 then µH = 1 +β(σ2�F

−σ�H �F )1+β .

• In general, ∂µH∂σ2�H

= −β(1−n)n1+β and∂µH

∂σ�H �F= β(1−n)(2n−1)1+β .


Optimal Consumption Sharesσ�H = σ�F = 1, σ�H�F = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

relative size of the home country

0.9

1

1.1

1.2

1.3

1.4

1.5Share in the world output

7H

7F

µH = 1 +βσ2�(n−1)(2n−1)

1+β , µF = 1 +βσ2�n(2n−1)

1+β .


Optimal Consumption Sharesσ�H = σ�F = 1, σ�H�F ∈ [−1, 1]

-1 -0.5 0 0.5 1

correlation coefficient

0.8

1

1.2

1.4

1.6

1.8

2n0=0.01

-1 -0.5 0 0.5 1


0.8

0.9

1

1.1

1.2

1.3

1.4n0=0.25

-1 -0.5 0 0.5 1


0

0.5

1

1.5

2n0=0.5

-1 -0.5 0 0.5 1


0.8

0.9

1

1.1

1.2

1.3

1.4n0=0.75

-1 -0.5 0 0.5 1


0.8

1

1.2

1.4

1.6

1.8

2n0=0.99

7H

7F

n7H

+(1-n)7F


What Do We Learn?

• Countries that are ex ante identical but for size, may end upwith a different level of (consumption) demand with full risksharing.

• Smaller countries gain. Intuitively, the equilibrium price of riskis close to the FA prices of the larger country.

• The gains are larger, the more negative the covariance ofoutput is.


Consumption LevelTwo-Factor Model

1. By construction, we have an example in which consumptionlevels in period 1 are equalized under financial autarky.

CH,1 = 1,CH,2 = 1− �H ,CH = 1.

2. as well as with income pooling:

CH,t = CF ,t where t = 1, 2.:

CH,1 = 1,CH,2 = (1− n�H − (1− n)�F ),CH = 1.

3. Under complete markets, however, because of the endogenousadjustment of income to the price of risk, Consumption is notequalized in period 1.

CH,1 = µH ,CH,2 = µH(1− n�H − (1− n)�F ),CH = µH .


WelfareTwo-Factor Model

Home loss under different risk sharing arrangements:

• FA: LFAH =β2σ

2�H

.

• CM: LCMH = −β(1− n)(n(σ�H�F − σ2�H )− (1− n)(σ�H�F −σ2�F )) +

β2 (n

2σ2�H + (1− n)2σ2�F + 2n(1− n)σ�H�F ).

• IP: LIPH =β2 (n

2σ2�H + (1− n)2σ2�F + 2n(1− n)σ�H�F ).


Welfare Rankingas a Function of Size and Volatility

Table 2: Ranking of CM, IP, FA for Home and Foreign countrydepending on home size n and σ2�H (σ�F = 1 and σ�H�F = 0)

n σ2�H0.01 < 0.98 (0.98, 99) (99, 199) > 1990.25 < 0.6 (0.6, 3) (3, 7) > 70.5 < 0.33 (0.33, 1) (1, 3) > 3

0.75 < 0.14 (0.14, 0.33) (0.33, 1.67) > 1.670.99 < 0.005 (0.005, 0.01) (0.01, 1.02) > 1.02

home CM � FA � IP CM � IP � FA IP � CM � FA IP � CM � FAforeign IP � CM � FA IP � CM � FA CM � IP � FA CM � FA � IP

• For 0.01 < σ2�H < 99 (given σ�F = 1) Home country prefers:• complete markets if its size is small (n=0.01).• income pooling if its size is large (n=.99).


Reversals in Quantitative ModelsSpurious Results

0 0.2 0.4 0.6 0.8 1


-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

perc

enta

ge o

f ste

ady

stat

e co

nsum

ptio

n

#10-4 Differences in Global Welfare Loss

0 0.2 0.4 0.6 0.8 1


-2

-1

0

1

2

3

4

5

perc

enta

ge o

f ste

ady

stat

e co

nsum

ptio

n

#10-4 Differences in Home Welfare Loss

Complete and Incomplete MarketsComplete Markets and Financial Autarky


Conclusions

• We reconsider the implications of risk sharing arrangementsamong countries that differ in size and business cycle.

• As the shadow price of future contingent output differs acrossborders, reforms that enhance risk-sharing lead to a change inrelative consumption vis-à-vis the status quo.

• Quantitative analysis: asymmetries among countries generallyincompatible with explicit/implicit equalization ofconsumption levels or income pooling.


Research Directions

• Interaction of financial frictions with other distortions.• In models with additional distortions, pecuniary externalities

may lower welfare—see e.g. Auray and Equyem (2014), whoallow for nominal rigidities but constrain monetary policy.

• Political economy of institutional reform and capital marketintegration.• large countries would prefer arrangement implementing income

pooling.• small countries may prefer complete markets, unless plagued

by cycles of large amplitude.

• Design of empirical assessment of risk sharing around times ofcapital market/institutional reforms.

.


Complete Markets Derivation (1)

maxU(CH) = u(CH,1) +S∑

s=1

βπ(s)u(CH,2(s))

s.t.CH,1 +S∑

s=1

p(s)

1 + rCH,2(s) = YH,1 +

S∑s=1

p(s)

1 + rYH,2(s)

Home and Foreign Euler equations:

CH,2(s) =

[π(s)β

(1 + r)

p(s)

] 1ρ

CH,1;CF ,2(s) =

[π(s)β

(1 + r)

p(s)

] 1ρ

CF ,1


Complete Markets Derivation (2)

Resource constraint:

nCH,1 + (1− n)CF ,1 = nYH,1 + (1− n)YF ,1 = YW ,1nCH,2(s) + (1− n)CF ,2(s) = nYH,2(s) + (1− n)YF ,2(s) = YW ,2(s)

Combining resource constraint and Euler equations:

YW ,2(s) =

[π(s)β

(1 + r)

p(s)

] 1ρ

YW ,1.

CH,1YW ,1

=CH,2(s)

YW ,2(s)=

CH,2(s′)

YW ,2(s ′)= µH .

where µi =

Yi,1

YρW ,1

+βE

(Yi,2

YρW ,2

)Yi,1

1−ρ+βE(Y 1−ρW ,2 )and i = H,F .

IntroductionModelEdgeworth BoxWelfare and ReversalsTwo-Factor ModelConclusions

Risk Sharing among Large and Small Countries · IntroductionModelEdgeworth BoxWelfare and...

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