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The following proofs assume permanent exclusion of credit markets after a default. Under this assumption the value function of default is independent of γL, and W. The quantitative analysis of the model generalizes the results to the case of temporary exclusion. We focus in b′

j < 0 (only in this situation is default possible), and the

equilibrium of the credit market implies ϑ′

j = −b′j > 0. More borrowing implies a

more negative b′

j.

Proposition 1 For any state of the world,S, as the risk aversion of the international investor increases, the emerging economies’ incentives to default increase.

Proof. Considering the case in which the government has not defaulted and assuming an interior solution for the allocation to the emerging economyj’s asset the first order condition of the investor’s problem is

φ ϑ′_j=E−qjvc cL ϑ′j

+βvc c′L ϑ′j

d′_j = 0.

Because the periodic utility of the international investor is of the CRRA type and γ₁L < γ₂L, then there exists a concave function ψ(·) such that v2 c;γ₂L =

ψ v1 c;γ₂L. If ϑ′_j,1 is the optimal allocation when γL = γ₁L, and ϑ′_j,2 is the op- timal allocation when γL _=γL

2 then it holds that

φ1 ϑ′_j,1=E−qjv1,c cL ϑ′_j,1+βv1,c c′_L ϑ′_j,1d′_j = 0. φ2 ϑ′j,2 =E−qjv2,c cL ϑ′j,2 +βv2,c c′L ϑ′j,2 d′_j = 0.

Using v2 c;γ₂L =ψ v1 c;γ₂L it is possible to define

where Υ(ϑ′_j) = ψ ′_{v c}′ L(ϑ′j ) ψ′_{v cL(ϑ}′ j ), Υ(ϑ ′ j)>0 and Υ′(ϑ′j)<0.30

The last inequality comes from the fact that both Υ(ϑ′_j) and ψ′ ϑ′_j are positive and decreasing. The inclusion of these functions in the previous equation implies that φ2 ϑ′_j,1 is lower than φ2 ϑ′_j,2 because Υ′_(ϑ′

j) and ψ′ ϑ′j

give little weight to the realizations of d_j′ = 1, and high weight to the realizations of d_j′ = 0. Therefore

φ2 ϑ′_j,2> φ2 ϑ′_j,1.

The concavity of VL(·) implies that given qj and the risk of default δj, φ ϑ′j

is a decreasing function and as a consequence ϑ′_j,2 < ϑ′_j,1 which in equilibrium implies

b′_j,2 > b′_j,1.

Then for any state of the world S, taking as given qj and (δj), a higherγL would

result in the investor allocating a lower proportion of her portfolio to each of the economies’ sovereign bonds. Therefore, when the investor is less risk averse there are financial contracts that are available to each emerging economy which are not available when the investor is more risk averse. Consequently given qj and δj then V_j,1C S;γ₁L ≥V_j,2C S;γ₂L

Because the utility of autarky for the emerging economy does not depend on γL_,

it is clear that if for someS, default is optimal for economyj whenγL_=γL

1,then for

the sameSdefault would be optimal when γL=γ₂L. Additionally, because incentives to default would be higher whenever γL = γ₂L, than when γL = γ₁L at equilibrium

δj S, bj′;γL₂> δj S, bj′;γ₁L, and thereforeqj S, bj′;γ₂L< qj S, bj′;γ₁L.

Proposition 2 Default sets are shrinking in the assets of the representative in- vestor. For all W1 < W2, if default is optimal for bj in some states yj, given W2 then default will be optimal for bj for the same states yj, given W1 therefore

30_{Taking as given the portfolio allocations of the investor to other emerging economies the deriva-} tive of Υ(ϑ′_j) is given by Υ′(ϑ′_j) = ψ′′[v(c′_L(ϑ′_j))]vc(c′L(ϑ′j)) ∂c′ L(ϑ′j) ∂ϑ′ j − ψ ′′_[v(c L(ϑ′j))]vc(cL(ϑ′j)) ∂cL(ϑ′j) ∂ϑ′ j Υ(ϑ ′ j) ψ′_{v c} L(ϑ′j ) <0.

Djbj |W2, ψ,{sk}J_k=1,k6=j⊆Djbj |W1, ψ,{sk}J_k=1,k6=j

Proof. Because the periodic utility of the international investor exhibit DARA

v W1, ϑ′j

is a concave transformation ofv W2, ϑ′j

so ifϑ′

j,1 is the optimal allocation

whenW =W1,andϑ′j,2is the optimal allocation whenW =W2,it is possible to define v1 ϑ′j,1 =v W1, ϑ′j,1 and v2 ϑ′j,2 =v W2, ϑ′j,2 , where v1 ϑ′j =ψ v2 ϑ′j . The first order conditions of the investor are

φ1 ϑ′j,1 = E−qjv1,c cL ϑ′j,1 +βv1,c c′L ϑ′j,1 dj′ = 0, φ2 ϑ′_j,2 = E−qjv2,c cL ϑ_j,2′ +βv2,c c′_L ϑ′_j,2dj′ = 0, and therefore φ1 ϑ′_j,2=Eψ′v2 ϑ′_j,2 −qjv2,c cL ϑ′_j,2+βΥ(ϑ′_j,2)v2,c c_L′ ϑ′_j,2dj′ <0.

Υ(ϑ′) is defined as before, and as before the inequality comes from the fact that Υ(ϑ′_j) and ψ′(ϑ′_j) are both positive and decreasing. Therefore φ1 ϑ′_j,2< φ1 ϑ′_j,1 .

Again the concavity of VL_{(·) implies that given q}

j and δj, φ ϑ′j

is a decreasing function, and as consequence ϑ′_j,2 > ϑ′_j,1 which in equilibrium implies b′_j,2 < b′_j,1.

Then for any S and taking as given qj and δj, a lower level of W would result in this agent allocating a lower proportion of her portfolio to the emerging economies’ sovereign bonds. Therefore, when the investor is more wealthy there are financial contracts that are available to the emerging economy that are not available when the investor is less wealthy. Consequently, given qj and δj then Vj,1C (S−W;W2) ≥ V_j,2C (S−W;W1). 31

Because the utility of autarky for the emerging economy does not depend on W, it is clear that if for somebj in some statesyj, default is optimal whenW =W2,then

for the same states yj default would be optimal when W =W1. Because incentives

to default would be higher whenever W = W1, than when W = W2 at equilibrium δj(S, bj′;W1′)> δ(S, bj′;W2′), and therefore q(s, bj′;W1′)< q(s, bj′;W2′).

31_S

Proposition 3 There is a wealth channel of contagion. Proposition 1 implies that if economy k in the investor’s portfolio defaults, then for economy j which is also in the investor’s portfolio, incentives to default increase.

Proof. If economyk defaults on her debts with the investor, the wealth of econ- omyk will be (W |dk= 0) =θT B′+

J P

m=1,m6=k

θ′_m,which is lower than her wealth if she decides not to default, which is (W |dk = 1) =θT B′ + PJ

m=1,m6=k θ′_m+θ′_k.Therefore V_jCyj, bj,{ym}m=1J ,{bm}Jm=1,(W |dk= 1) > V_jCyj, bj,{ym}m=1J ,{bm}Jm=1,(W |dk= 0)

which implies that emerging economy j′_{s incentives to default are larger when some} economy k which shares investors defaults.