This paper provides a necessary and sufficient condition for the existence of nonautarkic contract in a risk sharing model with two-sided lack of commitment. Verifying the condition takes just one Guassian elimination of a matrix

Tian, Guoqiang

Zhang, Yuzhe

English

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveWhen can we do better than autarky?Guoqiang Tian and Yuzhe ZhangTexas A&M University at College Station, Texas A&M University atCollege StationJanuary 2013Online at http://mpra.ub.uni-muenchen.de/44338/MPRA Paper No. 44338, posted 11. February 2013 02:24 UTCWhen can we do better than autarky?Guoqiang Tian1 and Yuzhe Zhang2Department of Economics, Texas A&M University, College Station, TX, 77843, United StatesAbstractThis paper provides a necessary and sucient condition for the existence of nonau-tarkic contract in a risk sharing model with two-sided lack of commitment. Verifyingthe condition takes just one Guassian elimination of a matrix.Keywords: Lack of commitment, Risk sharing, AutarkyJEL: D82, D861. IntroductionThe theory of contracting with two-sided lack of commitment has been appliedto study a wide range of economic issues, including international business cycles (cf.Kehoe and Perri (2002)), consumption inequality (cf. Kocherlakota (1996) and Ligonet al. (2002)), and wage contracts (cf. Thomas and Worrall (1988)). In this theory,a commonly made assumption is that some nonautarkic risk sharing arrangementis sustainable (in the sense that no one would leave the contract). To satisfy thisassumption, researchers focus on suciently patient economic agents, in which casea Folk-theorem argument shows that nearly any allocation is sustainable. Away fromthis extreme, a natural question is: Under what conditions does a nonautarkic andsustainable risk sharing arrangement exist?To answer this question, we study agents' incentives to participate in risk sharing.We linearize their utilities around autarkic endowment, which allows us to calculatein closed form the cost and the benet of participation. Hence the condition forparticipation is simply that the benet exceeds the cost. Besides answering theabove question, the analysis of the linearized model provides clear economic insightson agents' incentives that are dicult to identify in the original nonlinear model.1E-mail address: gtian@tamu.edu.2E-mail address: yuzhe-zhang@econmail.tamu.edu.Preprint submitted to Elsevier February 10, 20132. ModelThe model is similar to that in Ligon et al. (2002). There are two agents at timezero, with preferencesEhX1t=0tu(c1t )iand EhX1t=0tv(c2t )i;where cit (i = 1; 2) is agent i's consumption at time t,  2 (0; 1) is their commondiscount factor, and E is the expectation operator. Both agents are risk averse, i.e.,u00 < 0; v00 < 0. In each period t, agent i's income yi depends on the state of thenature s, which is drawn from a nite set f1; 2; :::; Sg and follows a Markov chain.Let  be the transition matrix [sr]Ss;r=1, where sr is the transition probability fromstate s to state r. We assume sr > 0 for all s and r to simplify the analysis.A risk sharing contract species for each t and each history ht  (s0; s1; :::; st) atransfer (ht) to be made from agent 1 to agent 2. Transfers can be negative. Neitheragent can commit; if one defaults, then both of them go to autarky (i.e., transfersare zero thereafter). Conditional on ht, the expected surplus of agent 1 over autarkyisU(ht)  u(y1(st)  (ht))  u(y1(st)) + E1Xj=t+1j t (u(y1(sj)  (hj))  u(y1(sj))) ;and the surplus of agent 2, V (ht), is dened similarly. A contract is sustainable ifU(ht)  0 and V (ht)  0, for all ht. All contracts to be discussed in this paper aresustainable.A sustainable contract is (constrained) ecient if for any given level of agent 1'ssurplus it provides more surplus to agent 2 than other sustainable contracts. Ligonet al. (2002) show that, if nonautarkic contracts exist, then an ecient contract ischaracterized as follows. There exist f Us > 0gSs=1 and agent 2's surplus functionsfVs() : [0; Us]! RgSs=1 such thatVs(Us) = maxs;fUrgSr=1v(y2(s) + s)  v(y2(s)) + SXr=1srVr(Ur) (1)subject to u(y1(s)  s)  u(y1(s)) + SXr=1srUr = Us; (2)Ur 2 [0; Ur]:The surplus function Vs(Us) decreases in Us and reaches zero at Us = Us.2.1. A linearized problemFollowing Thomas and Worrall (1988) and Ligon et al. (2002), this subsectionconsiders a model with utilities linearized around autarkic endowment. We show2below that the linearized model not only is analytically more tractable, it also of-fers clear intuition about the cost and the benet of participating in this long-termcontract. Fix f Us > 0gSs=1 in problem (1). Suppose agents' utilities in state s areu(y1(s))+u0(y1(s))(c1 y1(s)) and v(y2(s))+v0(y2(s))(c2 y2(s)). Agent 2's problemisLs(Us) = maxs;fUrgSr=1v0(y2(s))s + SXr=1srLr(Ur)subject to  u0(y1(s))s + SXr=1srUr = Us; Ur 2 [0; Ur]:Introducing cs  u0(y1(s))s, s  v0(y2(s))u0(y1(s)) , and Ar   Ur, we rewrite the above asLs(As) = maxcs;fArgSr=1scs + SXr=1srLr(Ar) (3)subject to cs + SXr=1srAr = As; (4)Ar 2 [  Ur; 0]:With a slight abuse of notation, we have used Ls(As) to denote Ls(Us). Without lossof generality, we assume that the ratio of marginal utilities s weakly increases in s.Problem (3) has the following interpretation. Both agent 1 and 2 have linearutilities and their consumptions are  cs and cs, respectively. Agent 2 is subjectto taste shocks fsgSs=1 while agent 1 is not. Because of the taste shocks, agent 2prefers consumption in states with high s while agent 1 is indierent. To facilitatetrade, agent 2 opens a \bank account" with agent 1, in which agent 2's asset holdingAs represents how much agent 1 owes agent 2. Noncommitment of agent 1 requiresAs  0 (i.e., agent 2 is in debt) at all times: positive As would obligate agent 1to repay and trigger his default. On the other hand, although agent 2 is in debt,he would not default as long as he can still benet from trading with agent 1. Tosee the benet, interpret (4) as agent 2's budget constraint. There are two channelsthrough which agent 2 can move consumptions from low-taste-shock states to high-taste-shock states: (1) he can reallocate assets among future states, holding moreassets in high-shock states; and (2) when the current taste shock is high, agent 2 canincrease his consumption through borrowing (i.e., holding less assets in the future).Calculating these benets is the key to understanding agent 2's default decision; thefollowing lemma does this in closed form.3Lemma 1. Ls(As) = Ls(0) + sAs, whereL(0) = (I   ) 1B U; (5)L(0) 0BBB@L1(0)L2(0)...LS(0)1CCCA ; B 0BBB@0 0    021(2   1) 0    0....... . ....S1(S   1) S2(S   2)    01CCCA ; U 0BBB@U1U2...US1CCCA :Proof: That Ls(As) is linear in As is because agent 2's utility function is linear. Tond out Ls(0), note that the optimal portfolio choice in problem (3) isAr =   Ur; if r < s;0; if r  s:Therefore, cs = As + Ps 1r=1 srUr, and the Bellman equation isLs(0) = s s 1Xr=1sr Ur!+ s 1Xr=1sr Lr(0)  r Ur+ SXr=ssrLr(0)= s 1Xr=1(s   r)sr Ur + SXr=1srLr(0):Solving the above linear system of equations yields (5). All elements in the matrix (I ) 1 are positive because (I ) 1 =P1t=0 tt.This and (5) imply that Ls(0)  0 for all s. If all ratios of marginal utilities areidentical (1 = s for all s), then autarky is the rst best outcome. In this case,B = 0 and L(0) = 0. If there are two states with dierent ratios of marginalutilities, then at least one element in B is positive. Then Ls(0) > 0 for all s becauseall elements in (I   ) 1 are positive.Remark 1. Ls(0) measures agent 2's benet from trading with agent 1. Becauseagent 2's initial asset holding is zero, his average consumption is zero too. Hence,the benet is purely from shifting consumptions from low to high taste-shock states.3. A necessary and sucient condition for nonautarkic contractThis section presents the main results of this paper. If nonautarkic contractsexist, then a necessary condition is as follows. Since the utility function v() is strictlyconcave, v0(y2(s))s > v(y2(s) + s)   v(y2(s)) holds whenever s 6= 0. Therefore, ifAs =   Us, then the surplus Ls(As) in the linearized problem (3) is higher thanVs( Us) in (1). That is,Ls(0)  s Us = Ls(As) > Vs( Us) = 0; for all s; (6)4or in matrix form M U < 0 (a vector is less than zero if all the elements are less thanzero), whereM 0BBB@1 0    00 2    0....... . ....0 0    S1CCCA  (I   ) 1B:Inequality (6) highlights the cost and the benet of the long-term relationship foragent 2. Recall that Ls(0) is agent 2's benet when he holds zero assets. If his initialasset holding is As =   Us in state s, then debt repayment costs him s Us units ofutility. This cost must be dominated by the benet if he participates in the contract.The above discussion shows (6) as a necessary condition for a nonautarkic con-tract. It turns out that (6) is also sucient. The intuition is as follows. Undercondition (6), a nonautarkic contract is sustainable with linear utilities. With non-linear utilities u() and v(), a risk sharing allocation would be sustainable if it mimicsthe contract with linear utilities in a small neighborhood around autarky, becauseu() and v() are well approximated by the linearized utilities in this neighborhood.Theorem 1. A nonautarkic contract exists if and only if there exists f Us > 0gSs=1to satisfy (6).Proof: Since necessity was already shown, this proof only shows suciency. Iff Us > 0gSs=1 satises (6), we construct a recursive nonautarkic contract as follows.In problem (1), choose agent 1's surplus asUr = Ur; if r < s;0; if r  s;where  > 0 is a small number to be determined later. Choose s to satisfy (2), theparticipation constraint of agent 1. Next we verify Vs( Us)  0;8s, the participationconstraint of agent 2. If  is small, (2) implies s    Us+Ps 1r=1 srUru0(y1(s)), which furtherimplies v(y2(s) + s)   v(y2(s))  s(  Us +Ps 1r=1 srUr). Hence for any  > 0,there is a suciently small  > 0 such thatv(y2(s) + s)  v(y2(s)) >   (s + ) Us +s 1Xr=1sr(s   ) Ur!:Algebra similar to that of Lemma 1 shows that agent 2's surplus is larger than(I   ) 10BBB@0 0    021(2   1   2) 0    0....... . ....S1(S   1   2) S2(S   2   2)    01CCCA0BBB@U1U2...US1CCCA  0BBB@(1 + ) U1(2 + ) U2...(S + ) US1CCCA :5Because the above vector is positive when  = 0 (recall (6)), it remains positive forsuciently small  due to continuity. Therefore, agent 2's participation constraint issatised and the constructed nonautarkic contract is sustainable. Remark 2. Because (I   ) 1 increases in , (5) implies that the benet Ls(0)increases in  as well. The cost s Us, however, is independent of . Hence it is easierto sustain a nonautarkic allocation when agents are more patient. In particular, thereis a unique cuto  such that nonautarkic contracts exist if and only if  > . Thisresult reinforces the notion in the literature that there is no nonautarkic contractwhen  is suciently small (cf. Proposition 2 (v) in Ligon et al. (2002)).Remark 3. Besides patience, other factors such as large variability of agents' in-comes also facilitate risk sharing. To see this, consider the two-state example inLjungqvist and Sargent (2004, Section 20.10), where the discount factor is 0:85 andthe utilities are u(c) = v(c) = c1 1  ,  = 1:1. Agent 1's income y1(s) is i.i.d. overtime and can be either 1  y or y 2 (0:5; 1) with equal probability. Agent 2's incomeis y2(s) = 1  y1(s). Compute M as2:201   1:202 01:631   1:632 2and M U < 0 becomes(2:201   1:202) U1 < 0; (7)(1:631   1:632) U1 + 2 U2 < 0: (8)If U2 =1:63(2 1) U122, then (8) is always satised. Inequality (7) and U1 > 0 require2:201 1:202 < 0, or y > 0:57. In other words, (nonautarkic) risk sharing exists inthis example if and only if agents' incomes exhibit enough variability. This result isconsistent with the nding in Krueger and Perri (2011) that public income insurancethrough progressive income taxation reduces private risk sharing.Although condition (6) is intuitive, it is not easily veriable as f UsgSs=1 is unknowna priori. Below we relate condition (6) (i.e., M U < 0 for some U > 0) to thedeterminants of the principal minors of M . We begin with an illustrative example.Example 1. Suppose S = 3 and 1 < 2 < 3. Denote the matrix (I   ) 1B as0@ a b 0c d 0e f 01A, where elements (a to f) are positive. The third column is zero becausethe third column of B is zero. Then M =0@ 1   a  b 0 c 2   d 0 e  f 31A. Since the thirdinequality,  e U1   f U2 + 3 U3 < 0, is always satised if U3 equals e U1+f U223 , we focuson nding U1 > 0, U2 > 0 to satisfy the rst two inequalities:(1   a) U1   b U2 < 0; (9) c U1 + (2   d) U2 < 0: (10)6If 1   a  0, then ( U1; U2) = (1; ) satises both inequalities if  > 0 is sucientlysmall. Similarly, if 2 d  0, then ( U1; U2) = (; 1) satises both inequalities. Whathappens if both 1   a and 2   d are positive? In this case, (9) and (10) can berewritten as1   ab<U2U1;U2U1<c2   d:A solution exists if and only if 1 ab< c2 d , i.e., the determinant of1   a  b c 2   dis negative.The following theorem generalizes the above conditions on determinants. Let Tbe the largest s such that s < S (i.e., r = S for all r > T ) and M(1 : s; 1 : s) bethe principal minor of M containing the rst s rows and s columns.Theorem 2. There exists f Us > 0gSs=1 to satisfy (6) if and only if either det(M(1 :T; 1 : T )) < 0 or det(M(1 : s; 1 : s))  0 for some s < T .3Proof: Necessity: Suppose there exists f Us > 0gSs=1 to satisfy (6). If M(1; 1)  0,then det(M(1 : 1; 1 : 1))  0 and the proof is nished. Otherwise, if M(1; 1) > 0,then use Guassian elimination to eliminate M(s; 1) for all s > 1. Denote the newmatrix as ~M . If ~M(2; 2)  0, then det(M(1 : 2; 1 : 2)) = det( ~M(1 : 2; 1 : 2))  0.(Note that the elementary row operations in Guassian elimination do not changethe determinant of M(1 : 2; 1 : 2)). Otherwise, if ~M(2; 2) > 0, then use Guassianelimination again. We can repeat this procedure as long as ~M(s; s) > 0 and nallyreach the elimination of UT . We show that ~M(T; T ) < 0 if ~M(s; s) > 0 for alls < T . Because all o-diagonal elements in M(1 : T; 1 : T ) are negative, theelementary row operations in Guassian elimination always add one row to anotherand hence preserve the signs of the inequalities in (6). The inequality in row Tbecomes ~M(T; T ) UT < 0, which implies ~M(T; T ) < 0 because UT > 0. Therefore,det(M(1 : T; 1 : T )) = det( ~M(1 : T; 1 : T )) < 0.Suciency: Construct a solution to (6) as follows. Carry out Guassian elim-ination until det(M(1 : s; 1 : s))  0 for the rst time. This means ~M(r; r) > 0for all r < s and ~M(s; s)  0. Let Us  1 and then recursively dene Ur  Psk=r+1 ~M(r;k) Uk~M(r;r)> 0 for r = s 1; s 2; :::; 1. ThatPsk=1 ~M(r; k) Uk  = 0; if r < s; 0; if r = s;implies thatPsk=1M(r; k)Uk= 0; if r < s; 0; if r = s: Dene Ur   > 0 for r = s+1; :::; T .Finally, we verifyPTk=1M(r; k)Uk < 0 for all r = 1; :::; T . If r  s, then the inequal-ity follows fromM(r; k) < 0 and Uk > 0 for k > s. If r > s, thenPTk=1M(r; k)Uk < 03Equivalently, autarky is the only sustainable contract if and only if det(M(1 : T; 1 : T ))  0and det(M(1 : s; 1 : s)) > 0 for all s < T .7holds when  > 0 is suciently small, because M(r; s) < 0 and Us = 1. Remark 4. As shown in the proof of Theorem 2, it takes only one Guassian elim-ination of M to verify the condition of this paper. This can be done easily in mostnumerical software.4. ConclusionThis paper establishes a necessary and sucient condition for the existence ofnonautarkic contracts in a model where two risk-averse agents face stochastic incomeand cannot commit. This condition is easy to verify as it boils down to computingdeterminants of matrices. The analysis of linearized utilities in the paper helps usunderstand the tradeos in an agent's participation decision.ReferencesKehoe, P., Perri, F., 2002. International business cycles with endogenous incompletemarkets. Econometrica 70 (3), 907{928.Kocherlakota, N., 1996. Implications of ecient risk sharing without commmitment.Review of Economic Studies 63 (4), 595{609.Krueger, D., Perri, F., 2011. Public versus private risk sharing. Journal of EconomicTheory 146, 920{956.Ligon, E., Thomas, J., Worrall, T., 2002. Informal insurance arrangements withlimited commitment: Theory and evidence from village economies. Review of Eco-nomic Studies 69, 209{44.Ljungqvist, L., Sargent, T., 2004. Recursive Macroeconomic Theory, Second Edition.The MIT Press, Cambridge, Massachusetts, USA.Thomas, J., Worrall, T., 1988. Self-enforcing wage contracts. Review of EconomicStudies 55, 541{554.8

When can we do better than autarky?

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
When can we do better than autarky?
Guoqiang Tian and Yuzhe Zhang
Texas A&M University at College Station, Texas A&M University at
College Station
January 2013
Online at https://mpra.ub.uni-muenchen.de/44338/
MPRA Paper No. 44338, posted 11. February 2013 02:24 UTC
When can we do better than autarky?
Guoqiang Tian1 and Yuzhe Zhang2
Department of Economics, Texas A&M University, College Station, TX, 77843, United States
Abstract
This paper provides a necessary and sucient condition for the existence of nonau-
tarkic contract in a risk sharing model with two-sided lack of commitment. Verifying
the condition takes just one Guassian elimination of a matrix.
Keywords: Lack of commitment, Risk sharing, Autarky
JEL: D82, D86
1. Introduction
The theory of contracting with two-sided lack of commitment has been applied
to study a wide range of economic issues, including international business cycles (cf.
Kehoe and Perri (2002)), consumption inequality (cf. Kocherlakota (1996) and Ligon
et al. (2002)), and wage contracts (cf. Thomas and Worrall (1988)). In this theory,
a commonly made assumption is that some nonautarkic risk sharing arrangement
is sustainable (in the sense that no one would leave the contract). To satisfy this
assumption, researchers focus on suciently patient economic agents, in which case
a Folk-theorem argument shows that nearly any allocation is sustainable. Away from
this extreme, a natural question is: Under what conditions does a nonautarkic and
sustainable risk sharing arrangement exist?
To answer this question, we study agents' incentives to participate in risk sharing.
We linearize their utilities around autarkic endowment, which allows us to calculate
in closed form the cost and the benet of participation. Hence the condition for
participation is simply that the benet exceeds the cost. Besides answering the
above question, the analysis of the linearized model provides clear economic insights
on agents' incentives that are dicult to identify in the original nonlinear model.
1E-mail address: gtian@tamu.edu.
2E-mail address: yuzhe-zhang@econmail.tamu.edu.
Preprint submitted to Elsevier February 10, 2013
2. Model
The model is similar to that in Ligon et al. (2002). There are two agents at time
zero, with preferences
E
hX1
t=0
tu(c1t )
i
and E
hX1
t=0
tv(c2t )
i
;
where cit (i = 1; 2) is agent i's consumption at time t,  2 (0; 1) is their common
discount factor, and E is the expectation operator. Both agents are risk averse, i.e.,
u00 < 0; v00 < 0. In each period t, agent i's income yi depends on the state of the
nature s, which is drawn from a nite set f1; 2; :::; Sg and follows a Markov chain.
Let  be the transition matrix [sr]
S
s;r=1, where sr is the transition probability from
state s to state r. We assume sr > 0 for all s and r to simplify the analysis.
A risk sharing contract species for each t and each history ht  (s0; s1; :::; st) a
transfer (ht) to be made from agent 1 to agent 2. Transfers can be negative. Neither
agent can commit; if one defaults, then both of them go to autarky (i.e., transfers
are zero thereafter). Conditional on ht, the expected surplus of agent 1 over autarky
is
U(ht)  u(y1(st)  (ht))  u(y1(st)) + E
1X
j=t+1
j t (u(y1(sj)  (hj))  u(y1(sj))) ;
and the surplus of agent 2, V (ht), is dened similarly. A contract is sustainable if
U(ht)  0 and V (ht)  0, for all ht. All contracts to be discussed in this paper are
sustainable.
A sustainable contract is (constrained) ecient if for any given level of agent 1's
surplus it provides more surplus to agent 2 than other sustainable contracts. Ligon
et al. (2002) show that, if nonautarkic contracts exist, then an ecient contract is
characterized as follows. There exist f Us > 0gSs=1 and agent 2's surplus functions
fVs() : [0; Us]! RgSs=1 such that
Vs(Us) = max
s;fUrgSr=1
v(y2(s) + s)  v(y2(s)) + 
SX
r=1
srVr(Ur) (1)
subject to u(y1(s)  s)  u(y1(s)) + 
SX
r=1
srUr = Us; (2)
Ur 2 [0; Ur]:
The surplus function Vs(Us) decreases in Us and reaches zero at Us = Us.
2.1. A linearized problem
Following Thomas and Worrall (1988) and Ligon et al. (2002), this subsection
considers a model with utilities linearized around autarkic endowment. We show
2
below that the linearized model not only is analytically more tractable, it also of-
fers clear intuition about the cost and the benet of participating in this long-term
contract. Fix f Us > 0gSs=1 in problem (1). Suppose agents' utilities in state s are
u(y1(s))+u
0(y1(s))(c1 y1(s)) and v(y2(s))+v0(y2(s))(c2 y2(s)). Agent 2's problem
is
Ls(Us) = max
s;fUrgSr=1
v0(y2(s))s + 
SX
r=1
srLr(Ur)
subject to  u0(y1(s))s + 
SX
r=1
srUr = Us; Ur 2 [0; Ur]:
Introducing cs  u0(y1(s))s, s  v0(y2(s))u0(y1(s)) , and Ar   Ur, we rewrite the above as
Ls(As) = max
cs;fArgSr=1
scs + 
SX
r=1
srLr(Ar) (3)
subject to cs + 
SX
r=1
srAr = As; (4)
Ar 2 [  Ur; 0]:
With a slight abuse of notation, we have used Ls(As) to denote Ls(Us). Without loss
of generality, we assume that the ratio of marginal utilities s weakly increases in s.
Problem (3) has the following interpretation. Both agent 1 and 2 have linear
utilities and their consumptions are  cs and cs, respectively. Agent 2 is subject
to taste shocks fsgSs=1 while agent 1 is not. Because of the taste shocks, agent 2
prefers consumption in states with high s while agent 1 is indierent. To facilitate
trade, agent 2 opens a \bank account" with agent 1, in which agent 2's asset holding
As represents how much agent 1 owes agent 2. Noncommitment of agent 1 requires
As  0 (i.e., agent 2 is in debt) at all times: positive As would obligate agent 1
to repay and trigger his default. On the other hand, although agent 2 is in debt,
he would not default as long as he can still benet from trading with agent 1. To
see the benet, interpret (4) as agent 2's budget constraint. There are two channels
through which agent 2 can move consumptions from low-taste-shock states to high-
taste-shock states: (1) he can reallocate assets among future states, holding more
assets in high-shock states; and (2) when the current taste shock is high, agent 2 can
increase his consumption through borrowing (i.e., holding less assets in the future).
Calculating these benets is the key to understanding agent 2's default decision; the
following lemma does this in closed form.
3
Lemma 1. Ls(As) = Ls(0) + sAs, where
L(0) = (I   ) 1B U; (5)
L(0) 
0BBB@
L1(0)
L2(0)
...
LS(0)
1CCCA ; B 
0BBB@
0 0    0
21(2   1) 0    0
...
...
. . .
...
S1(S   1) S2(S   2)    0
1CCCA ; U 
0BBB@
U1
U2
...
US
1CCCA :
Proof: That Ls(As) is linear in As is because agent 2's utility function is linear. To
nd out Ls(0), note that the optimal portfolio choice in problem (3) is
Ar =
   Ur; if r < s;
0; if r  s:
Therefore, cs = As + 
Ps 1
r=1 sr
Ur, and the Bellman equation is
Ls(0) = s
 

s 1X
r=1
sr Ur
!
+ 
s 1X
r=1
sr
 
Lr(0)  r Ur

+ 
SX
r=s
srLr(0)
= 
s 1X
r=1
(s   r)sr Ur + 
SX
r=1
srLr(0):
Solving the above linear system of equations yields (5). 
All elements in the matrix (I ) 1 are positive because (I ) 1 =P1t=0 tt.
This and (5) imply that Ls(0)  0 for all s. If all ratios of marginal utilities are
identical (1 = s for all s), then autarky is the rst best outcome. In this case,
B = 0 and L(0) = 0. If there are two states with dierent ratios of marginal
utilities, then at least one element in B is positive. Then Ls(0) > 0 for all s because
all elements in (I   ) 1 are positive.
Remark 1. Ls(0) measures agent 2's benet from trading with agent 1. Because
agent 2's initial asset holding is zero, his average consumption is zero too. Hence,
the benet is purely from shifting consumptions from low to high taste-shock states.
3. A necessary and sucient condition for nonautarkic contract
This section presents the main results of this paper. If nonautarkic contracts
exist, then a necessary condition is as follows. Since the utility function v() is strictly
concave, v0(y2(s))s > v(y2(s) + s)   v(y2(s)) holds whenever s 6= 0. Therefore, if
As =   Us, then the surplus Ls(As) in the linearized problem (3) is higher than
Vs( Us) in (1). That is,
Ls(0)  s Us = Ls(As) > Vs( Us) = 0; for all s; (6)
4
or in matrix form M U < 0 (a vector is less than zero if all the elements are less than
zero), where
M 
0BBB@
1 0    0
0 2    0
...
...
. . .
...
0 0    S
1CCCA  (I   ) 1B:
Inequality (6) highlights the cost and the benet of the long-term relationship for
agent 2. Recall that Ls(0) is agent 2's benet when he holds zero assets. If his initial
asset holding is As =   Us in state s, then debt repayment costs him s Us units of
utility. This cost must be dominated by the benet if he participates in the contract.
The above discussion shows (6) as a necessary condition for a nonautarkic con-
tract. It turns out that (6) is also sucient. The intuition is as follows. Under
condition (6), a nonautarkic contract is sustainable with linear utilities. With non-
linear utilities u() and v(), a risk sharing allocation would be sustainable if it mimics
the contract with linear utilities in a small neighborhood around autarky, because
u() and v() are well approximated by the linearized utilities in this neighborhood.
Theorem 1. A nonautarkic contract exists if and only if there exists f Us > 0gSs=1
to satisfy (6).
Proof: Since necessity was already shown, this proof only shows suciency. If
f Us > 0gSs=1 satises (6), we construct a recursive nonautarkic contract as follows.
In problem (1), choose agent 1's surplus as
Ur =

 Ur; if r < s;
0; if r  s;
where  > 0 is a small number to be determined later. Choose s to satisfy (2), the
participation constraint of agent 1. Next we verify Vs( Us)  0;8s, the participation
constraint of agent 2. If  is small, (2) implies s    Us+
Ps 1
r=1 sr
Ur
u0(y1(s))
, which further
implies v(y2(s) + s)   v(y2(s))  s(  Us +
Ps 1
r=1 sr
Ur). Hence for any  > 0,
there is a suciently small  > 0 such that
v(y2(s) + s)  v(y2(s)) > 
 
 (s + ) Us +
s 1X
r=1
sr(s   ) Ur
!
:
Algebra similar to that of Lemma 1 shows that agent 2's surplus is larger than
(I   ) 1
0BBB@
0 0    0
21(2   1   2) 0    0
...
...
. . .
...
S1(S   1   2) S2(S   2   2)    0
1CCCA
0BBB@
U1
U2
...
US
1CCCA  
0BBB@
(1 + ) U1
(2 + ) U2
...
(S + ) US
1CCCA :
5
Because the above vector is positive when  = 0 (recall (6)), it remains positive for
suciently small  due to continuity. Therefore, agent 2's participation constraint is
satised and the constructed nonautarkic contract is sustainable. 
Remark 2. Because (I   ) 1 increases in , (5) implies that the benet Ls(0)
increases in  as well. The cost s Us, however, is independent of . Hence it is easier
to sustain a nonautarkic allocation when agents are more patient. In particular, there
is a unique cuto  such that nonautarkic contracts exist if and only if  > . This
result reinforces the notion in the literature that there is no nonautarkic contract
when  is suciently small (cf. Proposition 2 (v) in Ligon et al. (2002)).
Remark 3. Besides patience, other factors such as large variability of agents' in-
comes also facilitate risk sharing. To see this, consider the two-state example in
Ljungqvist and Sargent (2004, Section 20.10), where the discount factor is 0:85 and
the utilities are u(c) = v(c) = c
1 
1  ,  = 1:1. Agent 1's income y1(s) is i.i.d. over
time and can be either 1  y or y 2 (0:5; 1) with equal probability. Agent 2's income
is y2(s) = 1  y1(s). Compute M as

2:201   1:202 0
1:631   1:632 2

and M U < 0 becomes
(2:201   1:202) U1 < 0; (7)
(1:631   1:632) U1 + 2 U2 < 0: (8)
If U2 =
1:63(2 1) U1
22
, then (8) is always satised. Inequality (7) and U1 > 0 require
2:201 1:202 < 0, or y > 0:57. In other words, (nonautarkic) risk sharing exists in
this example if and only if agents' incomes exhibit enough variability. This result is
consistent with the nding in Krueger and Perri (2011) that public income insurance
through progressive income taxation reduces private risk sharing.
Although condition (6) is intuitive, it is not easily veriable as f UsgSs=1 is unknown
a priori. Below we relate condition (6) (i.e., M U < 0 for some U > 0) to the
determinants of the principal minors of M . We begin with an illustrative example.
Example 1. Suppose S = 3 and 1 < 2 < 3. Denote the matrix (I   ) 1B as0@ a b 0c d 0
e f 0
1A, where elements (a to f) are positive. The third column is zero because
the third column of B is zero. Then M =
0@ 1   a  b 0 c 2   d 0
 e  f 3
1A. Since the third
inequality,  e U1   f U2 + 3 U3 < 0, is always satised if U3 equals e U1+f U223 , we focus
on nding U1 > 0, U2 > 0 to satisfy the rst two inequalities:
(1   a) U1   b U2 < 0; (9)
 c U1 + (2   d) U2 < 0: (10)
6
If 1   a  0, then ( U1; U2) = (1; ) satises both inequalities if  > 0 is suciently
small. Similarly, if 2 d  0, then ( U1; U2) = (; 1) satises both inequalities. What
happens if both 1   a and 2   d are positive? In this case, (9) and (10) can be
rewritten as
1   a
b
<
U2
U1
;
U2
U1
<
c
2   d:
A solution exists if and only if 1 a
b
< c
2 d , i.e., the determinant of

1   a  b
 c 2   d

is negative.
The following theorem generalizes the above conditions on determinants. Let T
be the largest s such that s < S (i.e., r = S for all r > T ) and M(1 : s; 1 : s) be
the principal minor of M containing the rst s rows and s columns.
Theorem 2. There exists f Us > 0gSs=1 to satisfy (6) if and only if either det(M(1 :
T; 1 : T )) < 0 or det(M(1 : s; 1 : s))  0 for some s < T .3
Proof: Necessity: Suppose there exists f Us > 0gSs=1 to satisfy (6). If M(1; 1)  0,
then det(M(1 : 1; 1 : 1))  0 and the proof is nished. Otherwise, if M(1; 1) > 0,
then use Guassian elimination to eliminate M(s; 1) for all s > 1. Denote the new
matrix as ~M . If ~M(2; 2)  0, then det(M(1 : 2; 1 : 2)) = det( ~M(1 : 2; 1 : 2))  0.
(Note that the elementary row operations in Guassian elimination do not change
the determinant of M(1 : 2; 1 : 2)). Otherwise, if ~M(2; 2) > 0, then use Guassian
elimination again. We can repeat this procedure as long as ~M(s; s) > 0 and nally
reach the elimination of UT . We show that ~M(T; T ) < 0 if ~M(s; s) > 0 for all
s < T . Because all o-diagonal elements in M(1 : T; 1 : T ) are negative, the
elementary row operations in Guassian elimination always add one row to another
and hence preserve the signs of the inequalities in (6). The inequality in row T
becomes ~M(T; T ) UT < 0, which implies ~M(T; T ) < 0 because UT > 0. Therefore,
det(M(1 : T; 1 : T )) = det( ~M(1 : T; 1 : T )) < 0.
Suciency: Construct a solution to (6) as follows. Carry out Guassian elim-
ination until det(M(1 : s; 1 : s))  0 for the rst time. This means ~M(r; r) > 0
for all r < s and ~M(s; s)  0. Let Us  1 and then recursively dene Ur 
 Psk=r+1 ~M(r;k) Uk
~M(r;r)
> 0 for r = s 1; s 2; :::; 1. ThatPsk=1 ~M(r; k) Uk  = 0; if r < s; 0; if r = s;
implies that
Ps
k=1M(r; k)
Uk

= 0; if r < s;
 0; if r = s: Dene Ur   > 0 for r = s+1; :::; T .
Finally, we verify
PT
k=1M(r; k)
Uk < 0 for all r = 1; :::; T . If r  s, then the inequal-
ity follows fromM(r; k) < 0 and Uk > 0 for k > s. If r > s, then
PT
k=1M(r; k)
Uk < 0
3Equivalently, autarky is the only sustainable contract if and only if det(M(1 : T; 1 : T ))  0
and det(M(1 : s; 1 : s)) > 0 for all s < T .
7
holds when  > 0 is suciently small, because M(r; s) < 0 and Us = 1. 
Remark 4. As shown in the proof of Theorem 2, it takes only one Guassian elim-
ination of M to verify the condition of this paper. This can be done easily in most
numerical software.
4. Conclusion
This paper establishes a necessary and sucient condition for the existence of
nonautarkic contracts in a model where two risk-averse agents face stochastic income
and cannot commit. This condition is easy to verify as it boils down to computing
determinants of matrices. The analysis of linearized utilities in the paper helps us
understand the tradeos in an agent's participation decision.
References
Kehoe, P., Perri, F., 2002. International business cycles with endogenous incomplete
markets. Econometrica 70 (3), 907{928.
Kocherlakota, N., 1996. Implications of ecient risk sharing without commmitment.
Review of Economic Studies 63 (4), 595{609.
Krueger, D., Perri, F., 2011. Public versus private risk sharing. Journal of Economic
Theory 146, 920{956.
Ligon, E., Thomas, J., Worrall, T., 2002. Informal insurance arrangements with
limited commitment: Theory and evidence from village economies. Review of Eco-
nomic Studies 69, 209{44.
Ljungqvist, L., Sargent, T., 2004. Recursive Macroeconomic Theory, Second Edition.
The MIT Press, Cambridge, Massachusetts, USA.
Thomas, J., Worrall, T., 1988. Self-enforcing wage contracts. Review of Economic
Studies 55, 541{554.
8


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When can we do better than autarky?

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