Altruism and Risk Sharing in Networkspareto.uab.cat/pdf/seminars/micro/2017/Barcelona2017.pdf ·...

Altruism and Risk Sharing in Networks

Renaud Bourlès, Yann Bramoullé & Eduardo Perez-Richet

Aix-Marseille School of Economics & Sciences Po

September 2017

Introduction

I Informal transfers are still prevalent in our 21st centuryeconomies.

I Transfers in cash, kind, or time between individuals andhouseholds and which are not market transactions.

I Key stylized facts:I Quantitatively large, even in high-income countries.I Interact with the business cycle and with public transfers.I Flow through social networks.I Generate (not too) ineffi cient informal insurance.

Introduction: quantitatively large

I In low-income countries, remittances often on par with theformal economy.

I In 2009, Lebanon: 22% of GDP; Jordan: 16%; Philippines12%, Worldbank (2011).

I Greater than financial aid and close to foreign directinvestment, Yang (JEP 2011).

I Remarkably stable following the crisis. Between 2008 and2009, dropped by 5% while FDI dropped by 40%.

I Also important in high-income countries.I In France, family transfers increased from 2% of GDP beforethe crisis to 4% after, Le Monde (2014).

I In the US, interhousehold transfers estimated at 1.2% of GDPin 2003, National Transfer Accounts (2011).

Introduction: interact with public transfers

I Large empirical evidence on crowding out.I Increase in public transfers may decrease private transfers.Reduces the impact of public interventions.

I In a study on the Philippines, between 30% to 80% ofcrowding out for those in the lowest income quintile, Cox,Hansen & Jimenez (JPubE 2004).

I “attempts to aid the poor could be thwarted by privateresponses, which leak benefits to richer households in the formof lighter burdens of support for less fortunate kin.”

I Recent evidence that transfers to the poor indirectly benefitsocially connected households.

I Angelucci & De Giorgi (AER 2009), Angelucci, De Giorgi &Rasul (WP 2012).

Introduction: social networks

I Expanding empirical literature studying bilateral transfers.I Dercon & De Weerdt (JDE 2006); Fafchamps & Lund (JDE2003); Fafchamps & Gubert (JDE 2007).

I Recent work by Arun Chandrasekhar, Pascaline Dupas andothers.

I Even in small rural communities, it is not true that everyone ishelping everyone else.

I Rather, informal transfers are structured through socialnetworks: close and distant relatives, friends, neighbors.

Introduction: ineffi cient insurance

I Following Towsend (ECA 1994), large empirical literature thattests effi cient insurance on consumption data.

I Typically rejected at the village level. Still, does not seem tooineffi cient: consumption little affected by individual shocks.

I Often interpreted as a sign that informal risk-sharing worksquite well.

I Mazzocco & Shaini (AER 2012) develop new versions of thetests to account for preference heterogeneity.

I On data on rural India, reject effi cient insurance at the villagelevel but not at the subcaste level.

Introduction: motives

I Why do people give? Three main explanations.I Exchange: Mutually beneficial informal insurance contract.I Altruism: People give to others they care about.I Social pressure: People give because they feel obliged to.

I Identifying the motives empirically is challenging.I Large literature following Cox (JPE 1987). Recent studiesbased on clever experiments.

I Evidence that the three motives are at work, although maybeacross different types of ties and circumstances.

Introduction: theory

I Growing theoretical literature rationalizing these four facts.I Models of informal transfers in networks.

I Links as social collateral, Ambrus, Mobius & Szeidl (AER2014).

I Links have values constraining how much money can flowthrough them.

I Characterize Pareto-constrained risk-sharing arrangements.

I Local information constraints, Ambrus, Gao & Milan (WP2017).

I How to reach these Pareto-constrained arrangements?

Introduction: theory

I Altruism in networks, Bourlès, Bramoullé & Perez-Richet(ECA 2017).

I Altruism à la Becker, structured through a network.I Characterize Nash equilibria of the game of transfers, fornon-stochastic incomes.

I Even in the absence of risk, altruism generates informalredistribution.

I Not true under informal insurance contracts.

Introduction: altruism in networks

I Advances the economics of altruism.I Following Becker (JPE 1974) and Barro (JPE 1974).I Large literature but irrealistic structures: Small groups ofcompletely connected agents or linear dynasties.

I However, family ties form complex networks.I Well-known from human genealogy.I Argued early on by Bernheim & Bagwell (JPE 1988) but hadnot been explored by economists.

Introduction: altruism and risk sharing

I In this new paper, we study the risk sharing implications ofaltruism networks.

I Incomes are stochastic, transfers conditional on incomes as inBBP (2017).

I Becker (JPE 1974)’s early intuition: “The head’s concernabout the welfare of other members provides each, includingthe head, with some insurance against disasters.”

I Never studied in a network context.

I We find that altruism networks have a strong impact on risk.

Introduction: altruism and risk sharing

I Informal insurance tends to be better if the network has loweraverage path length.

I We characterize what happens for small shocks.I Partially insured by endogenous risk-sharing communities.

I We show that large shocks tend to be well-insured.I Throughout, we contrast outcomes under altruism and undersocial collateral.

I We uncover complex structural effects.I A new link may decrease or increase the risk faced by others.

Model: informal transfers

I Agent i has income y0i and may give tij ≥ 0 to agent j .I Matrix T = (tij ) represents the network of informal transfers.

I Consumption yi is equal to

yi = y0i −∑jtij +∑

k

tki

I Aggregate income is conserved: ∑i yi = ∑i y0i .

Model: altruism in networks

I Agents care about others’well-being:

vi (y) = ui (yi ) +∑j

αijuj (yj )

I Coeffi cient αij ∈ [0, 1] measures the strength of the altruisticlink from i to j .

I Network of altruism (αij ) describing the structure of socialpreferences.

I i may care about j but not about j’s friends. Interests of agiver and a receiver may be misaligned.


I Noncooperative game: Agents makes transfers to maximizetheir altruistic utilities.

I Transfers by an agent depend on transfers made by others.

I T is a Nash equilibrium iff (1) tij > 0⇒ u′i (yi ) = αiju′j (yj )and (2) ∀i , j , u′i (yi ) ≥ αiju′j (yj ).

I Under CARA ui (y) = −e−Ay ,(1) tij > 0⇒ yi = yj + (− ln(αij ))/A and (2)∀i , j , yi ≤ yj + (− ln(αij ))/A.

I An agent does not let the consumption of a poorer friendbecome too much lower than his own.


Theorem (BBP 2017) A Nash equilibrium always exists.Equilibrium consumption y is unique. Generically in α, there is aunique Nash equilibrium T.

I Emergence of transfer intermediaries in equilibrium.I Give to poorer friends part of the money received from richerfriends.

I Shocks propagate in the altruism network. Example on theline.

Equilibria on the line

u CARA, links have same strength: ‐ln(α)/A=1

10 010 10

8 69 731 6

Altruism and risk

I Suppose now that incomes are stochastic.I How do altruistic transfers in networks affect risk?

I Interplay of two countervailing forces: (1) Conditional on y0−i ,altruistic transfers reduce risk.

Proposition Conditional on y0−i , yi − E (yi ) Second-OrderStochastically Dominates y0i − E (y0i ).

I Proof: yi is weakly increasing in y0i but yi − y0i is weaklydecreasing in y0i

I Changes in transfers made or received caused by changes inown income tend to smooth own consumption.

Altruism and risk

I (2) Conditional on y0i , altruistic transfers tend to increase risk.Bear part of the income risk of others.

I To give examples, useful benchmark with no redistribution inexpectation.

Proposition Under CARA, symmetric incomes and undirected tiesαij = αji , ∀i , Eyi = Ey0i .

I Proof: If T equilibrium transfers for E (y0) + ε, reversetransfers Tt equilibrium for E (y0)− ε.

Altruism and risk

I Simple example with 2 agents and iid binary shocks.I When i has non-stochastic income, yi more risky than y0iwhile yj less risky.

I When both agents have stochastic incomes, consumption lessrisky for both.

I Here, the first force dominates the second.

++ -+ +- --

Altruism and risk sharing: 2 agents

++ -+ +- --

+ -

+ -

Model: effi cient insurance

Definition Informal transfers yield effi cient insurance if ∃λ � 0such that consumption y solves

maxy ∑

iλiEui (yi )

subject to ∑i yi = ∑i y0i .

I Classical notion underlying empirical analysis followingTownsend (1994).

I With common utilities and equal Pareto weights, leads toequal income sharing ∀i , yi = y0.

I In general, u′i (yi )/u′j (yj ) = λj/λi in every state of the world.

I Under CARA, yi = y0 + ln(λi )/A.

Effi cient insurance and perfect altruism

Proposition Equilibrium transfers induce effi cient insurance withequal Pareto weights if any two agents are indirectly connectedthrough a path of altruistic links of strength 1.

I Proof: If αij = 1, u′i (yi ) ≥ u′j (yj ). Path from i to j and pathfrom j to i yield u′i (yi ) = u

′j (yj ).

I Perfect altruism between pairs aggregate up into effi cientinsurance.

I Equal income sharing between pairs leads to overall equalsharing in a connected network.

I Even when sparse: stars, circle or line.I Agents act as if they were altruistic towards their friends’friends.

Insurance and imperfect altruism

I How far does society get from effi cient insurance whenaltruism is imperfect?

I Following Ambrus, Mobius & Szeidl, introduce distance fromequal income sharing: DISP(y) = E 1n ∑i |yi − y0 |.

I Define cij = − ln(αij ) virtual cost of link ij and cij =least-cost of paths connecting i to j .

I If the network is binary and αij ∈ {0, α}, cij = cdij where dij= network distance between i and j .

I Let d = 1n(n−1) ∑i 6=j dij be the average path length.

I cij = extension of network distance to weighted networks.


PropositionUnder CARA and undirected ties αij = αji ,

DISP(y) ≤ 1A1n2 ∑

i ,jcij

I For binary networks αij ∈ {0, α},

DISP(y) ≤ − ln(α)A

n(n− 1)n2

d

I Proof: Sum of equilibrium inequalitiesyi ≤ yj + (− ln(αij ))/A.


I Under CARA, informal insurance induced by altruism tends tobe better when average path length is lower.

I Contrasts to the role played by expansiveness under socialcollateral.

I Informal insurance subject to small-world effects underaltruism but not under social collateral.

I A few links between communities may have a strong impact.

I Bound extends to directed networks, other measures ofdistance and other utilities.

I Dispersion of path lengths around the mean may also matter.

Effi cient insurance and small shocks

I Are there other circumstances where effi cient insurance isachieved?

I Perhaps surprisingly, the answer is yes.I We next characterize what happens for small shocks.

I From equilibrium T, define the graph of transfers G as gij = 1if tij > 0 and gij = 0 otherwise.

I Consider the (weak) components of G.I Generically in α and y0, small shocks y0 + ε yield the samegraph of transfers.


I If i indirectly connected to j in G, define cij = ∑s εis is+1cis is+1where εis is+1 > 0 if tis is+1 > 0 and εis is+1 < 0 if tis is+1 < 0 inpath from i to j .

I Net virtual cost of indirect connection from i to j .I Example. If αij = αji ∈ {0, α}, cij = directed distance betweeni and j in G.

I Satisfies cji = −cji and triangular equality cij + cjk = cik .

Directed cost

i

j

k

l

cil= cij+ cjk‐ ckl


Theorem

(1) Generically in α and y0: Consider small shocks around y0.Then, equilibrium transfers induce effi cient insurance withincomponents of G for Pareto weights λi = exp( 1nC ∑j∈C cij ).

(2) Generically in α: Suppose that society is partitioned incommunities, equilibrium transfers induce effi cient insurance withincommunities and y0i is continuous with positive density over itssupport. Then the graph of transfers is constant across incomerealizations and its components are the communities.


I Idea of the proof: (1).I Assemble Nash conditions on positive transfers. Yieldsln(u′i )− ln(u′j ) = −cij for any i ,j in a component of G.

I We then check that ln(λi )− ln(λj ) = cij . In addition,∑i ln(λi ) = 0.

I Idea of the proof: (2).I Consider open set with an equilibrium transfer graph G. By(1), λ(G) = λ.

I Generically in α, the mapping G→ λ(G) is injective.


I Components of G constitute endogenous risk sharingcommunities.

I Extends Theorem 3 in BBP (2017) on income pooling to risksharing. Characterizes functions fi .

I Within components, equilibrium behavior equivalent to aplanner’s program.

I Pareto weight λi = exp(1nC ∑j∈C cij ) increasing in average net

distance between i and other members of the component.I Tends to be larger for givers, smaller for receivers.


I Quality of insurance depends on the connectedness of G.I Small shocks are effi ciently insured if G is connected.

I Happens if α is connected and i very rich or very poor.I Money then flows from i to everyone or from everyone to i .I Adjustements in altruistic transfers generate effi cient insurance.


I Small shocks are not insured if G is empty.I Happens with similar incomes y0i ≈ y0j .I Agents then bear all the risk associated with small shocks.

I In general, the quality of insurance depends on thecomponents’sizes.

I Under CARA, yi = y0C +1A ln(λi ). With iid incomes,

Var(yi ) =1nCVar(y0i ).

Large shocks

I Consider opposite benchmark, when an agent is subject tolarge shocks.

I Recall, cij = least cost of indirect connection between i and j .

Proposition Under CARA and α connected, for any y0−i thereexist YH > YL such that:

y0i ≥ YH ⇒ yi = y0 + 1A1n ∑j cij

y0i ≤ YL ⇒ yi = y0 − 1A1n ∑j cji .

I If i’s income is very low or very high and α connected,everyone involved in informal transfers.

I Except in the middle range, yi varies linearly in y0i with slope1/n. As with effi cient insurance.

Large shocks

Proposition Consider the model of social collateral with tij ≤ κij .Under CARA, for any y0−i there exist YH > YL such that

y0i ≥ YH ⇒ yi = y0i − cH and y0i ≤ YL ⇒ yi = y0i + cL

I If shocks are large, informal transfers saturate links’maximumcapacity.

I Except in the middle range, yi varies linearly in y0i with slope1. As without insurance.

Large shocks

I Say that shocks on i become arbitrarily large if for any ∆,prob(|y0i − E (y0i )| ≤ ∆)→ 0.

I DISP(yi ) = E |yi − y0 |.

Corollary Suppose u CARA and α connected. When shocks on ibecome arbitrarily large, DISP(yi ) stays bounded under altruismbut becomes arbitrarily large under social collateral.

I Large shocks well-covered under altruism but not under socialcollateral.

I Around equal incomes, the opposite happens for small shocks.

Network structure and insurance

I Impact of new link on risk faced by others?I Consider i connected to j and form new link between j and k.Impact on i?

I Two countervailing forces. k is a source of indirect support.Could reduce the risk faced by i .

I k is also a competitor for j’s help. Could increase i’s risk.

I From numerical simulations, we see the two situationsemerging and in different circumstances.

I Simulations with u CARA, y0 normally distributed iidN(10, σ = 3) and − ln(α)/A = 1.

I 100, 000 runs per network to recover the full consumptiondistribution.

Network structure and insurance

I A new link between separate communities helps.I By contrast, new link in connected nets often increasesconsumption variance of indirect neighbors.

I Line to circle: variance reduction for agents close to the newlink and variance increase for agents far from it.

I We also find that more central agents tend to have lowervariance.

I With correlated incomes, a new link between separatecommunities may increase risk.

Link ij reduces Var(yk) Link ij increases Var(yk)

4.72 4.72 9

3.55 3.553.16

4.72 4.72 4.72 4.72

3.13 3.132.56 2.56

9

3.16

3.55 3.55

2.38

2.92

2.922.92

2.382.92

2.92

2.921.89

2.50

2.50

2.50

2.50

9

3.16

3.55 3.55

3.20

3.20 3.20

2.05

2.24 2.23

2.05 2.23

2.56 2.56

3.133.13

2.602.60

2.60 2.60

2.20

A new link may decrease variance for some indirect neighbors and increase it for others

2.99

2.99 2.33

2.33 2.13

2.20

2.20 2.20

2.20 2.20

A link between communities can increase risk for indirect neighbors

0 3 6 0 3 6

6 3 0 6 3 0

2 3 4 2 3 4

4 3 2 4 3 2

1.5 2.5 3.5 2.5 3.5 4.5

4.5 3.5 2.5 3.5 2.5 1.5

1 0 1 1 0 1 2.25 0.25 0.25 0.25 0.25 2.25

y0

y

Var(y)

Conclusion: summary

I We analyze the risk sharing properties of altruism networks.I Informal insurance tends to be better when the average pathlength is lower.

I Small shocks effi ciently insured if the graph of transfers isconnected.

I Large shocks tend to be well-insured.I Rich structural effects.

I Outcomes quite distinct from the model of social collateral.

Conclusion: future research

I Endogenous networks: Bramoullé & Kranton (JEBO 2007),Ambrus, Chandrasekhar & Elliott (WP 2015).

I Risk sharing affects marriage, Rosenzweig & Stark (JPE 1989).

I Altruism and incentives in networks.I Altruism may help solve moral hazard problems, Alger &Weibull (AER 2010).

I Structural estimations of models of informal transfers innetworks.

I Or even simply detailed empirical investigation ofintermediation. If i is linked with j who is linked with k, howdoes a shock on k affects i?

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