Consumption Risksharing in Social Networks

Transcription

1 onsumption Risksharing in Social Networks Attila Ambrus Adam Szeidl Markus Mobius Harvard University U-Berkeley Harvard University April 2007 PRELIMINARY

2 Introduction In many societies, formal insurance markets are missing or imperfect ) agents use informal arrangements - Such arrangements often take place in the social network. This paper builds a model of informal insurance in the social network. { onnections generate value which is used as social collateral to enforce informal contracts. Our model yields results consistent with stylized facts about limited risksharing in developing countries.

3 Plan of the Talk 1. Theory: Informal insurance in social networks. 2. Model analysis: A) The limits to risksharing; B) onstrained ecient arrangements and local sharing.

4 Related literature Bloch, Genicot and Ray (2006): haracterize networks that are stable under certain risksharing arrangements. Bramoulle and Kranton (2005): Explore risksharing without enforcement constraints. Mobius and Szeidl (2006): Borrowing in a social network. Wilson (1968): Risksharing in syndicates. ochrane (1991), Townsend (1994), Udry (1994): Limits to full risksharing in the data.

5 The benefits of risksharing Adam Ilya

6 The benefits of risksharing Adam Ilya 50% ω Nature 50% ω +1-1

7 The benefits of risksharing Adam Ilya Nature 50% 50% ω ons: 0 0 ω +1-1 ons: 0 0

8 The benefits of risksharing Adam Ilya Nature 50% 50% ω -1 $1 +1 ons: 0 0 ω +1-1 $1 ons: 0 0

9 Risksharing with informal enforcement Adam c Ilya 50% ω Nature 50% ω +1-1

10 Risksharing with informal enforcement Key condition: Transfer link capacity Adam c Ilya 50% ω Nature 50% ω +1-1

11 Risksharing with informal enforcement Key condition: transfer link capacity Adam c Ilya Suppose c<1, then. 50% ω Nature 50% ω +1-1

12 Risksharing with informal enforcement Key condition: transfer link capacity Adam c Ilya Suppose c<1, then. 50% ω -1 $c +1 Nature 50% ω +1 $c -1

13 Risksharing with informal enforcement Key condition: transfer link capacity Adam c Ilya Suppose c<1, then. Nature 50% 50% ω -1 $c +1 ons: -(1-c) 1-c ω +1 $c -1 ons: 1-c -(1-c)

14 General model onsider N agents organized in exogenous social network G = (W; L) : A link represents friendship or business relationship between parties. Strength of relationships measured by non-negative capacity c : W W! R. { Assume that c (i; j) = c (j; i) and that c (i; j) = 0 if (i; j) =2 L.

15 Endowments and transfers A state of the world is characterized by a realization of agents' endowments: fe i g i2w { Endowments have a commonly known prior distribution. { All endowment realizations in product set of supports have positive probability. To share risk, agents ex ante agree on a set of transfers t ij for all i, j 2 W and for each state of the world. { Here t ij is the net transfer to be paid by i to j. { By denition, t ij = t ji.

16 ontract enforcement After endowment realization, agents are expected to make promised transfers. If agent i fails to make promised transfer t ij, he loses (i; j) link. These links disappear exogenously: friendly feelings no longer exist when somebody breaks a promise. { an be microfounded: if j fails to receive t ij payment, he concludes that j is no longer a friend, and stops interacting with him.

17 Incentive compatible transfers Transfer arrangement is incentive compatible if agents make promised transfers for all realizations of uncertainty. { Better to pay transfer than to lose friend. Arrangement is incentive compatible i capacity con- Proposition: straint holds: t ij c (i; j) : Promised transfers should never exceed friendship value.

18 2. Model Analysis: Limits to risksharing Key point: one network statistic seems to govern limits to risksharing in many environments. For any subset of agents F, dene perimeter-to-area ratio of F as where c [F ] = P i2f, j =2F a [F ] = c [F ] jf j c (i; j) is the total capacity leaving F. aptures ability of set in ooading shocks relative to number of agents in set. For cubes of side n in the k-dimensional grid, a [F ] = O n 1.

19 Limits to risksharing Let i be the standard deviation of e i, and let = min [ i ]. Proposition. If a [F ] < for some F with jf j N=2, then no I insurance arrangement is unconstrained Pareto-optimal, and there is an agent in F who is not fully insured. { If perimeter/area is not big enough, rst best insurance cannot be obtained. Intuition: shocks accumulate over agents in F ; if perimeter is small, some accumulated shocks cannot fully leave F.

20 50% +σ Infinite line network e i -σ 50%

21 50% +σ Infinite line network e i -σ 50% F

22 50% +σ Infinite line network e i -σ 50% F e 0 =+σ

23 50% +σ Infinite line network e i -σ 50% F e 0 =+σ e 1 =+σ

24 50% +σ Infinite line network e i -σ 50% F e 0 =+σ e 1 =+σ e 2 =+σ e 3 =+σ

25 50% +σ Infinite line network e i 50% -σ F σ c[f] F e 0 =+σ e 1 =+σ e 2 =+σ e 3 =+σ

26 Limits to risksharing: partial converse Let e i be coin ips assuming values and and assume with equal probabilities, (1=N) X e i = 0: Proposition. If a [F ] for all F with jf j N=2, then there exists an I arrangement implementing equal sharing. { Sharp characterization of when is rst-best implementable. { Proof uses max ow - min cut result from computer science.

27 Imperfect risksharing: a measure of dispersion When Pareto-optimality fails, what degree of risksharing can be achieved? First need to develop a measure of the degree of risksharing. Dene cross-sectional dispersion of consumption as square root of expected cross-sectional variance: DISP = E 1 N X (xi x) 2 1=2 : { With i.i.d. shocks and no sharing (autarky), DISP = ; { In equal sharing DISP = 0.

28 Imperfect risksharing: a characterization result Basic point: Risksharing in network can reduce DISP by perimeter/area ratio. Let e i be coin ips as above =) DISP in autarky is. Proposition [Perimeter/area ratio and degree of risksharing] (i) If a [F ] k for all F with jf j N=2, then there exists an I allocation where DISP = k. (ii) There exist networks and endowment distributions where this is sharp: DISP k for all I arrangements.

29 Imperfect risksharing: proof outline In any endowment realization, there are N=2 people with + and N=2 people with. { Need to transfer $k from + guys to guys. This is a ow problem =) feasible if min cut desired ow. { Perimeter/area bounds ensure that min cut is always big enough.

30 Limits to risksharing: correlated shocks Suppose that agents are partitioned into disjoint sets (villages) F m, m = 1, 2,...,M. Agents experience both village-specic and idiosyncratic shocks: e i = s m + " i where s m is common shock for village m and " is i.i.d. To measure risksharing across villages, regress village per capita consumption on village-specic shock: y m = m + m s m + m : In full risksharing, m = O (1=N) 0; with no risksharing, m = 1.

31 orrelated shocks continued High risksharing across villages means close to zero; low risksharing means close to 1. Proposition. We have 1 a [F m ] sm m 1 + a [F m] sm : When the perimeter/area ratio a [F m ] of a village is small, is close to one. Village-specic shock cannot leave F m through thin perimeter.

32 Limits to risksharing: summary Perimeter-to-area ratio relative to governs limits to risksharing: 1. When ratio small, full risksharing cannot be achieved; 2. Ratio governs magnitude of cross-sectional cons dispersion; 3. Ratio limits degree to which village-specic shocks can be shared. onsistent with evidence on limited risksharing: { ochrane (1991): Large shocks (long unemployment spell, illness) are not fully insured. { Townsend (1994): More insurance within than across villages.

33 2B onstrained ecient allocations an we learn more by focusing on \second best" arrangements? A risksharing arrangement is constrained ecient if it is Pareto-optimal subject to the I constraints. Our approach: 1. Second best arrangements are equivalent to solution of social planner's problem with a set of weights. 2. Use FO of planner's problem to characterize second-best allocations.

34 FO of planner's problem Assume all planner weights are equal. FO with no I constraints would be: U 0 i = U 0 j for all i, j. FO with I constraints: if U 0 i < U 0 j then t ij = c (i; j). Intuition: 1. If marginal utility is not equalized, planner's objective can be improved by transferring a small amount from i to j. 2. This is not possible =) I constraint must hold with equality.

35 Endogenous risksharing \islands" Fix endowment realization fe i g i2w : FO can be used to partition network into connected components, such that 1. Within a component, marginal utility is equalized; 2. Across components, marginal utility diers and connecting links operate at full capacity. Shock are fully shared within an \island" but imperfectly shared across islands.

36 Example: ircle network onsider N agents organized in a circle w/ all link capacities = c. Suppose that e i are identically distributed: where K > 4c. e i = ( K with prob p 0 with prob 1 p { K represents adverse shocks such as unemployment or illness. Focus on ecient arrangement with equal planner weights. { If i and j are in the same equivalence class =) x i = x j.

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39 c c c c c c

40 c c c 3c/5 c c/5 c/3 c/5 c/3 3c/5 c c

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42 Local sharing Island result suggests that socially closer agents share more risk with each other. To formalize this, x realization e = (e i ) and dene \bad shock" e 0 such that e 0 i < e i and e 0 j = e j for all j 6= i. { e 0 is an idiosyncratic negative shock to i. Measure impact of shock on agent j by marginal utility cost e 0 MU j = U i 0 Ui 0 (e) where larger MU j corresponds to a more painful shock.

43 Marginal utility cost of shock (MU) Utility cost for agent with shock (MU i ) Utility cost for direct friend (MU l ) Utility cost for indirect friend (MU j ) Size of shock measured by utility cost to i

44 onclusion We developed a model of informal risksharing in social networks, where enforcement is provided by the \collateral value" of social links. Perimeter/area ratio governs limits to risksharing in many environments. In second best arrangements, agents organize in endogenous groups to share endowment shocks. { Socially close agents share more risk with each other.