Risk Sharing Between Households
Marcel Fafchamps (2008)
From Handbook of Social Economics
Editors Jess Behabib, Alberto Bisin & Matthew O. Jackson
1
Notes by Team Grossman (Spring 2015)
Meagan Madden and Mavzuna Turaeva
1 Visualization of family “cliques” in Costa Rica (Freeman 2000)
Table of Contents
1. Introduction ............................................................................................................................................... 3
2. Efficient Risk Sharing ............................................................................................................................... 3
Social Planner’s Problem ...................................................................................................................... 4
Model with “Taste Shifters” ................................................................................................................. 6
3. Forms of Risk Sharing .............................................................................................................................. 6
Channels of Risk Sharing ...................................................................................................................... 6
Types of Risk Sharing ........................................................................................................................... 6
4. The Motives for Risk Sharing ................................................................................................................... 7
4.1 Self-Interest and Repeated Interaction ................................................................................................ 8
Rational Self Interest ............................................................................................................................. 8
Model with Repeated Games ................................................................................................................ 8
Stationary vs. Non-stationary strategies .............................................................................................. 10
4.2 Intrinsic Motivation .......................................................................................................................... 11
Model for Altruism (Foster & Rosenzweig 2001) .............................................................................. 11
4.3 Evidence on Motives ......................................................................................................................... 13
Perfect Enforcement ............................................................................................................................ 13
Persistent Shocks ................................................................................................................................ 13
Altruism vs. Social Norms .................................................................................................................. 14
Nature of Transfers ............................................................................................................................. 14
5. Risk Sharing Groups and Networks ........................................................................................................ 14
5.1 Groups ............................................................................................................................................... 14
5.2 Networks ........................................................................................................................................... 15
6. Extensions ............................................................................................................................................... 17
7. References ............................................................................................................................................... 18
1. Introduction Risk is an inherent part of our day-to-day lives: death, medical illness, unexpected car repairs, and
unemployment are examples of uncertain events or “shocks” that could happen at any time. Risk-averse
individuals seek to insure against negative shocks. “Insurance” takes many forms, including formal and
informal mechanisms for pooling risks across individuals. For example, you can purchase a medical
insurance plan to protect yourself from negative health shocks or life insurance to protect your family if
you die. Some countries also provide social insurance benefits, such as disability or unemployment
insurance.
In some contexts (e.g. developing countries) the formal mechanisms and institutions to insure risk are
weak. Families and individuals in developing countries often face greater risk and have less wealth to
absorb the effects of negative shocks. As an example, think about subsistence farmers all over the world
who depend directly on their harvest each season; weather, disease, and natural disasters are very real and
omnipresent risks. Naturally, individuals/groups have found alternative strategies for reducing one’s
exposure to risk.
One informal mechanism is for an individual to spread his/her risk across time periods; for example, an
individual may save income (preserve food) today in anticipation of future shocks. Another type of
informal insurance is risk sharing within a household, where members of a household insure each other.
Say that a husband and wife are both employed; they can insure the other if he/she becomes unemployed
(or take over responsibilities on the farm). Alternatively, multiple households may insure each other. For
example, family groups often look to extended relatives for help during negative shocks, such as helping
during the critical parts of the growing season is someone is injured or ill. This is an example of risk
sharing between households, and this type of informal insurance is the topic of Marcel Fafchamps’
chapter in Handbook of Social Economics (2008).
Fafchamps summarizes the literature on risk sharing between households, including theoretical models
and empirical evidence. He defines this research as “the realm of informal arrangements, transcending
household boundaries without taking the form of market transactions.” Most of the models and empirical
evidence he presents are related to development contexts.
In our handout, we will introduce ideas of risk sharing across households, including:
Background on risk and efficient risk sharing
Motivations for sharing risk between households
Types of risk sharing between households
Formal models of risk sharing between households
For several models briefly described by Fafchamps–altruism, repeated game settings–we refer to the
original papers and further develop those models here.
2. Efficient Risk Sharing Fafchamps starts with simplified model to explain the conditions for efficient risk sharing. Assume that
households operate in a closed-economy, earning income and spending income on consumption goods. In
any period 𝑡, any household 𝑗 may suffer an income shock. If income decreases then consumption will
decrease, unless 𝑗 had pooled risk with other households to insure their consumption or found some other
way to recover loss of wealth (e.g. selling assets). In this model, households have no assets and aggregate
income must be consumed within each period.
Societal welfare is maximized when there is efficient risk sharing, so that household 𝑖’s consumption dips
after a shock only to the extent that the total income in the society relative. Intuitively, this means that
when household 𝑖 suffers a shock, all the remaining households pool resources to take care of household
𝑖. In the next period, if household 𝑗 suffers a shock, he will similarly be insured. See the mathematical
formulation below; the other models we present are developed through extensions and variations of this
simple model of efficient risk sharing.
Variable Description
𝐽 Number of households
𝑦𝑡𝑗 Income for household 𝑗 at time 𝑡
𝑐𝑡𝑗 Consumption for household 𝑗 at time 𝑡
𝑈𝑗(∙) Utility function for household 𝑗
𝑠 State of nature
𝛽 Discount factor
𝜋(𝑠𝜏𝑡) Probability of state of the world 𝑠𝜏 at time 𝑡
𝜔𝑗 Welfare weights
𝜆(𝑠𝜏𝑡) Lagrange multiplier associated with each feasibility constraint
𝑦𝑡𝑎 Aggregate income
𝑐𝑡𝑎 Aggregate consumption
𝑏𝑡𝑗
Expenses associated with a “taste shifting” shock, such that household require
more consumption in order to manage the shock (e.g. health shocks)
Social Planner’s Problem
The intertemporal expected utility for a single household 𝑗 is the discounted utility from income. Income
is a function of the state of the world (𝑠) which formalizes the effect of “shocks”, and each state has an
associated probability 𝜋:
𝐸𝑈 = ∑ 𝛽𝑡 ∑ 𝜋(𝑠𝜏𝑡)𝑈𝑗[𝑐𝑡𝑗(𝑠𝜏𝑡)]𝑆
𝜏=1∞𝑡=0 (1)
The social planner wants to maximize the discounted utility of all households, where each household is
assigned a welfare weight, 𝜔𝑗:
max ∑ 𝜔𝑗𝐽𝑗=1 ∑ 𝛽𝑡∞
𝑡=0 ∑ 𝜋(𝑠𝜏𝑡)𝑈𝑗[𝑐𝑡𝑗(𝑠𝜏𝑡)]𝑆
𝜏=1 (2)
s.t. ∑ 𝑐𝑡𝑗(𝑠𝜏𝑡)𝑗 = ∑ 𝑦𝑡
𝑗(𝑠𝜏𝑡)𝑗 (3)
The objective function (2) is maximized subject to the binding “feasibility constraint” (3) that aggregate
consumption (LHS) is equal to aggregate income (RHS). The feasibility constraint holds for each state of
the world 𝑠𝜏𝑡, so the maximization problem requires a Lagrange multiplier 𝜆(𝑠𝜏𝑡) for each state 𝑠. Take
the derivative of the Lagrangian with respect to 𝑐𝑡𝑗(𝑠𝜏𝑡) to find the FOC:
𝜔𝑖𝛽𝑡𝜋(𝑠𝜏𝑡)𝑈𝑖′[𝑐𝑡
𝑖(𝑠𝜏𝑡)] − 𝜆(𝑠𝜏𝑡) = 0 (4)
Then, we can find the conditions for Pareto Efficiency where the ratio of welfare between two households
𝑖 and 𝑗 is unchanged for every state of the world (𝑠, 𝑠′ ∈ 𝑆). Intuitively, this means that no household can
do better without making others worse off, given a set of random shocks (i.e. the state of the world).
𝜔𝑗
𝜔𝑖 = 𝜆(𝑠𝜏𝑡)/𝛽𝑡𝜋(𝑠𝜏𝑡)𝑈𝑗
′[𝑐𝑡𝑗(𝑠𝜏𝑡)]
𝜆(𝑠𝜏𝑡)/𝛽𝑡𝜋(𝑠𝜏𝑡)𝑈𝑖′[𝑐𝑡
𝑖(𝑠𝜏𝑡)]=
𝑈𝑖′(𝑠)
𝑈𝑗′(𝑠)
=𝑈𝑖
′(𝑠′)
𝑈𝑗′(𝑠′)
(5)
Equation (5) shows that the ranking of welfare weights 𝜔𝑗/𝜔𝑖 does not change with the state of the world
(𝑠, 𝑠′ ∈ 𝑆). This condition implies efficient risk sharing.
As an example, let’s assume that state 𝑠 entails a minor income shock for household 𝑖 (e.g. a small section
of crops on the farm fails). Let’s also assume that state 𝑠′ delivers a huge negative income shock to
household 𝑗; for example, the household head with most responsibilities on the farm dies. Without risk
sharing you would expect household 𝑖 is worse off in state of the world 𝑠, and household 𝑗 even more
worse off in state 𝑠′. However, if households maintain their relative welfare position in both states, then
they are perfectly pooling their risk and transferring wealth to the households experiencing shocks, minor
or major.
Such allocation can be achieved through conditional transfers 𝜏𝑠𝑖 = 𝑦𝑠
𝑖 − 𝑐𝑠𝑖 , for all households 𝑖 and
states of the world 𝑠. Transfers are subject to constraint such that net transfers over all households equal
zero (∑ 𝜏𝑠𝑖
𝑖 ∈𝑁 = 0, ∀ 𝑠). By the first welfare theorem, the perfect competitive equilibrium solution to the
social planner’s problem achieves Pareto efficiency in risk sharing. For an interior solution to (5), we
assume that households are risk-averse. Otherwise, they would not seek insurance. For example, assume
the concave form of 𝑈(𝑐) = log(𝑐), so 𝑈′(𝑐) = 1/𝑐 and Pareto Efficiency is of the form:
𝜔𝑗
𝜔𝑖 =𝑐𝑡
𝑗(𝑠)
𝑐𝑡𝑖(𝑠)
=𝑐𝑡+𝑣
𝑗(𝑠′)
𝑐𝑡+𝑣𝑖 (𝑠′)
(6)
Equation (6) shows that the ratio of consumption is fixed across states of nature (𝑠, 𝑠′ ∈ 𝑆) and across time
(𝑡, 𝑡 + 𝑣 ∈ Time). This implies that the level of consumption for a single household matters only in
relation to the level of all other households. In other words, 𝑐𝑡𝑗(𝑠𝜏𝑡) only varies with aggregate income.
This leads to an empirical test (7) where the reduced form equation for household consumption is a
function of aggregate consumption (𝑐𝑡𝑎) and aggregate income (𝑦𝑡
𝑗).
𝑐𝑡𝑗
= 𝛼1(𝑐𝑡𝑎) + 𝛼2(𝑦𝑡
𝑗) + 𝑣𝑗 + 𝑒𝑡
𝑗 (7)
The regression equation contains underlying household effects 𝑣𝑗 as well as idiosyncratic errors 𝑒𝑡𝑗. Some
underlying household characteristics which affect consumption include family structure, preferences for
consumption, location, and cost of living. We can estimate using first-differencing (8) to eliminate the
time-invariant household effects (𝑣𝑗):
𝑐𝑡+1𝑗
− 𝑐𝑡𝑗
= 𝛼1(𝑐𝑡+1𝑎 − 𝑐𝑡
𝑎) + 𝛼2(𝑦𝑡+1𝑗
− 𝑦𝑡𝑗) + 𝑒𝑡
𝑗 (8)
If 𝛼1=1, households efficiently share risk so household income varies with aggregate income/aggregate
consumption; if 𝛼2 = 0, households are able smooth consumption such that consumption today is not
affected by change in income due to a negative shock. Empirical evidence in development contexts
suggests that risk sharing occurs but is not completely efficient. Any risk sharing agreements, implicit or
explicit, are difficult to enforce, so efficiency is not often achieved. This is especially true for persistent
shocks that last for multiple periods 𝑡, such as unemployment.
Model with “Taste Shifters”
Fafchamps develops an additional model to show that “risk sharing” (measured by 𝛼1), and “consumption
smoothing” (measured by 𝛼2) are distinct concepts. There are certain types of shocks called “taste
shifters” which require greater compensation to manage the shock. For example, health shocks require
increased consumption in the form of healthcare expenditures. Equation (9) shows the expected utility for
household 𝑗 when there are 𝑏𝑡𝑗 expenditures on health for state 𝑠𝜏𝑡:
𝐸𝑈 = ∑ 𝛽𝑡 ∑ 𝜋(𝑠𝜏𝑡)𝑈𝑗[𝑐𝑡𝑗(𝑠𝜏𝑡) − 𝑏𝑡
𝑗(𝑠𝜏𝑡)]𝑆𝜏=1
∞𝑡=0 (9)
With health shocks, households require greater income than in other periods in order to cover expenses
𝑏𝑡𝑗. If you observe empirically that households maintain consumption, despite healthcare expenses, then
consumption smoothing holds but efficient risk sharing does not (i.e. 𝛼1 ≠ 1 and 𝛼2 = 0).
3. Forms of Risk Sharing Given the reduced form (8), we can imagine scenarios where there is no risk sharing agreement between
households, but the coefficients 𝛼1 and 𝛼2 indicate risk sharing:
If households have assets and liquidate those assets during shocks, they can smooth consumption
(𝛼2 = 0) without any risk sharing.
Say shocks are widespread enough that output decreases and prices go up. Then the real interest
rate rises because prices today (with the shock) are high relative to the next period. Even if the
income for a single household was unaffected by the widespread shock, there consumption will
decrease because they are incentive by the high interest rate to spend more and consume less. In
this case, individual consumption 𝑐𝑡𝑗 is correlated with aggregate output 𝑐𝑡
𝑎 (𝛼1 = 1) even without
mutual insurance.
Since it is difficult to identify the effects of explicit mutual insurance via reduced for (8), researchers also
investigate the channels for risk sharing and the types of risk sharing arrangements.
Channels of Risk Sharing
Ex post – help households to manage shocks after they have happened
Ex ante – allow households to reduce their exposure to risk before shocks happen
Types of Risk Sharing
Type Channel Description
Informal loans Ex post
A friendly loan that typically carries no interest and no
specified date for repayment; repayment contingent on
income
Gifts & transfers Ex post Altruistic transfers provided with no expectation of
repayment
Labor pooling Ex ante / Ex post
Households pool labor to complete all farming work and
insure against health shocks; pooling takes many forms
including labor gangs and rotating arrangements (can be
set up ex ante) as well as ex post pooling to help a
household in need of timely work (e.g. harvest season)
Funeral society Ex ante Share risk of high funeral costs by pooling money to pay
for funerals
Fostering children Ex post Extended family take care of children after death of a
parent, or so children can attend school
Family networks Ex post
Rely on close networks to cope with shocks; for example,
seek shelter during a natural disasters or outbreak of
violence, or seek information on employment
Business partnerships Ex ante
Pool resources and share commercial risk to start a new
business; partnerships often develop from close family or
friend networks
Since relatives are in long-term relationships exchange between family members rarely takes a form of
defined market transaction. Mutual assistance between family takes the form of gifts, while between
friends and distant relatives take the form of “loans”. Evidence of preferential hiring and higher wages
paid to relatives; A much more common form of family involvement in the business as unpaid help.
4. The Motives for Risk Sharing As shown by the social planner’s maximization problem, sharing risk is Pareto Efficient (5), so
households gain from pooling risk.
For idiosyncratic risks (i.e. shocks are random and independently distributed across households), risk-
averse households should pool risk to insure each other. As an example, consider a farming community
where the young males do most of the farm work. All households face the risk of losing their sons due to
random deaths (assuming there are not community-wide disasters that kill off multiple young laborers). In
this case, households should pool labor to compensate for any random deaths.
For covariate risk (i.e. shocks are correlated across households), the households with low degrees of risk
aversion should insure the households with greater risk aversion. For example, consider a wealthy
entrepreneur who is less averse to risk from natural disaster than the subsistence farmers in a community.
Mutual gains arise if the business man insures the farmers. In models of formal insurance contracts, we
typically assume risk-neutrality of insurance companies.
In theory, the legal mechanisms can enforce risk sharing agreements. In practice, legal enforcement is
difficult because it requires a reliable judicial system. This also requires complete, written contracts which
are costly or infeasible. Additionally, in development contexts, the transactions are too tiny to justify
judicial action. Therefore, a Pareto Efficient equilibrium may not be achieved if the costs of enforcement
are too difficult through formal channels.
Therefore, households often rely on other mechanisms to enforce contracts and sustain participation in
risk sharing agreements:
Emotional responses through altruism, guilt, and shame
Long-term strategic interactions
4.1 Self-Interest and Repeated Interaction
Rational Self Interest
Economists have mostly focused on rational self-interest mechanism to enforce informal contracts. For
example, through repeated interaction and quid pro quo attitude (“I help you today because I expect you
to help me tomorrow”) contracts can be self-enforcing. Experimental evidence supports quid pro quo
behavior in repeated game scenarios.
Model with Repeated Games
Variables Description
𝑐𝑠𝑖 = 𝑦𝑠
𝑖 Guaranteed consumption level for household 𝑖 for state of nature 𝑠 if she
continues to participate in the informal risk sharing arrangement
𝑦𝑡𝑖 Income for household 𝑖 in period 𝑡
𝛽𝑡 Discount factor
𝑦𝑡𝑖 Autarky level of consumption 𝑖 reverts to if she reneges on her obligations.
𝑠 State of the world/available information
𝑡 → ∞ Infinite time horizon
𝐸𝑠 Expectations based on 𝑠 (𝐸𝑠 > 0, when 𝑖 is risk averse)
Sequential game theory is used to demonstrate that an implicit agreement to share risk can be sustained
through repeated interactions and promises to assist others can be self-enforcing. The simple model of
risk sharing is expanded to infinite periods. Self-enforcement is determined by voluntary participation of
households. This depends on the difference in expected total utility from defecting on an informal
agreement vs. participating in the agreement:
𝑈𝑖(𝑦𝑠𝑖) − 𝑈𝑖(𝑐𝑠
𝑖) ≤ ∑ 𝛽𝑡𝐸𝑠(𝑈𝑖(𝑐𝑡𝑖) − 𝑈𝑖(𝑦𝑡
𝑖)∞𝑡=𝑠+1 ) (10)
The sequential game model assumes:
Non-satiation & risk aversion (𝑈′(∙) > 0, 𝑈′′(∙) ≤ 0 ∀𝑖 ; 𝑈′′(∙) < 0 for some 𝑖)
All 𝑖′s face uncertain income streams
Similar time preference for all households (𝛽𝑖𝑡 = 𝛽𝑗
𝑡)
Repeated interaction is required (𝑡 > 1)
The LHS of equation (10) represents the short-term gain from defecting. It is positive whenever
household 𝑖 transfers income to other households (𝜏𝑠𝑖 < 0). The RHS represents the total expected utility
from continued participation. For risk-averse households 𝑖 the RHS is positive so that they gain by
sharing risk. If the LHS is less than the RHS, then households will continue to participate in the risk
sharing agreement and contract is enforced. For an informal risk sharing arrangement to be self-enforcing,
inequality (10) must hold in all states of the world 𝑠, for all time periods 𝑡, and all households 𝑖.
Further conditions for a contract to be self-enforcing over time include immobility of households, infinite
horizon, and constant conditional probability of interaction. Stable rural communities and fishing
communities are one application. Often, social obligations and stigma “are inherited within dynastic
households”, so the next generation joins the household and 𝑡 → ∞.
Another condition is patience across periods (high 𝛽). With more impatient households, risk sharing is
less likely to be self-enforcing. Figure 1 shows the shape of the equilibrium payoff set for various values
of 𝛽. The figure illustrates a well- known result that the set of equilibria for repeated games shrinks as
agents get more impatient (Fafchamps 1998).
Size of transfers
The self-enforcement through self-interest sets an upper limit on the amount of transfer, 𝜏𝑠𝑖 < 0 (ignoring
altruism). Egalitarian risk sharing cannot be achieved if some households are approximately risk neutral
and able to self-ensure. In such cases there is an asymmetric risk sharing in the form of “patronage”
whereby the rich (typically, less risk averse) ensure the poor (typically more risk averse)
Persistent vs. Temporary Shocks
The voluntary participation constraint (10) is easier to satisfy for short-lived shocks and harder to satisfy
for persistent shocks, such health shocks due to chronic illness or permanent disability. Fafchamps
presents a model with that includes risk sharing transfers (positive or negative) as well as negative health
transfers.
Variable Description
𝐹 Expected utility when health shocks are considered
ℎ𝑠𝑖 > 0 Health shock shared by agents 𝑖 and 𝑗
𝜏𝑠𝑖 Transfer from 𝑖 to 𝑗
𝜏𝑠𝑚𝑎𝑥 = 𝜏𝑠
𝑚𝑎𝑥(𝐸𝑠[𝐹]) Maximum transfer from 𝑖 to 𝑗 (i.e. the larger 𝐸𝑠[𝐹], the larger 𝜏𝑠𝑚𝑎𝑥)
𝜏𝑠𝑖 < 𝜏𝑠
𝑚𝑎𝑥 Sustainable transfer
The expected utility from participating in the risk sharing agreement is defined by 𝐹 and voluntary
participation is determined by the inequality (12):
𝐹 ≡ ∑ 𝛽𝑡(𝑈𝑖(𝑦𝑠𝑖 − 𝜏𝑠
𝑖 − ℎ𝑠𝑖 ) − 𝑈𝑖(𝑦𝑡
𝑖 − ℎ𝑠𝑖 )∞
𝑡=𝑠+1 ) (11)
𝑈𝑖(𝑦𝑠𝑖 − ℎ𝑠
𝑖 ) − 𝑈𝑖(𝑦𝑠𝑖 − 𝜏𝑠
𝑖 − ℎ𝑠𝑖 ) ≤ 𝐸𝑠[𝐹] (12)
The highest amount of transfer that still satisfies voluntary participation is denoted 𝜏𝑠𝑚𝑎𝑥. This max value
is lower in the presence of persistent health shocks because your expected utility of the risk-sharing
agreement (RHS) decreases since you will be paying for health expenses in the foreseeable future:
discounted 𝐸𝑠[𝐹] ⇒ 𝜏𝑠𝑚𝑎𝑥 ⇒ 𝜏𝑠
𝑖 ↓ ↓ ↓
Notice that as health issues persist in subsequent periods necessitating transfer from agent 𝑖 to the
member of the group, agent 𝑖 becomes increasingly anxious about expected gains from participation in
the game, thus becomes more present-oriented. The longer the duration of health shock to the other agent,
the less beneficial is for agent 𝑖 to participation. This may result either in agent 𝑖 discontinuing
participation in the risk sharing arrangement or in expulsion of agent 𝑗 form the group. This shows that
risk sharing based on anticipated reciprocity is less able to ensure against persistent shocks than against
transitory shocks. The logic: someone affected by permanent shock is less able to reciprocate in the
future, thus becomes a less valuable member of the risk sharing arrangement.
Stationary vs. Non-stationary strategies
Fafchamps then makes the distinction between gifts which are memory-less transfers and quasi-credit
where repayment is expected. It is called quasi-credit be the repayment conditions are much less strict
than formal credit mechanisms: (1) repayment is renegotiated to reflect shocks affecting the lender and
borrower, (2) no interest charged, and (3) access to such credit is need-based (Platteau and Abraham
1987).
When transfer 𝜏𝑠𝑖 takes the form of quasi-credit instead of a gift, the voluntary participation constraint
(10) is easier to satisfy and risk sharing is more likely.
Gift giving describes the “stationary” strategies, i.e. strategies that depend on the current state of nature
and not on the previous transfers. Quasi-credit, on the other hand, is an example of “non-stationary”
transfers that preserves the “memory” of the past. These types of transfers can be formalized by the
models below.
Gift-Giving Model
𝑈(𝑦𝑠′𝑖 ) − 𝑈𝑖(𝑐𝑠′
𝑖 ) ≤𝛽
1−𝛽𝐸[𝑈𝑖(𝑐𝑠
𝑖) − 𝑈𝑖(𝑦𝑠𝑖)] (13)
Here, the LHS gives the utility without gift-giving and implied insurance. The RHS gives the discounted
expected utility from getting or giving a gift. Because of the “memory-less” assumption of gift-giving, the
expected future transfer on the RHS is much lower than other forms of risk sharing.
Quasi-Credit Model
𝑈𝑖(𝑦𝑠′𝑖 ) − 𝑈𝑖(𝑐𝑠′
𝑖 ) ≤ ∑ 𝛽𝑡𝐸𝑈𝑖(𝑐𝑡𝑖) −∞
𝑡=𝑠+1𝛽
1−𝛽𝐸𝑈𝑖(𝑦𝑠
𝑖) (14)
In this model, you expect a much larger payoff by participating in the quasi-credit scheme
(RHS). Therefore, the voluntary participation is easier to satisfy, and participation in the risk
sharing should be more likely over time.
4.2 Intrinsic Motivation
The literature has also begun to investigate the role of emotions to enforce informal contracts. For
example, the anticipated emotions from satisfying or failing a contract can motivate participation and
contractual obligation:
Guilt- negative emotion from failing to meet your obligation
Self-righteousness- positive emotion from succeeding to meet your obligation & conforming to
group expectations
Shame- negative emotion through disapproval of others; requires that others know of your
wrong-doing
Altruism- capacity to feel good by helping others
The psychology and evolutionary biology literature suggest that these human emotions have an important
role in societal relationships and evolutionary history. Similar to the “survival of the fittest” idea, those
families that are most “fit” are most likely to pass on their genetic material. One idea from evolutionary
biology is that families that are more altruistic towards each other are more “fit”. This is supported by the
findings of Dawkins (1989) that individuals with more shared genetic material are more altruistic toward
each other. Foster and Rosenzweig (2008) formally model altruism in a risk sharing model with imperfect
commitment, which we describe below.
Model for Altruism (Foster & Rosenzweig 2001)
Variable Description
𝑠𝑡 State of nature 𝑠 in period 𝑡
𝑦𝑖(𝑠𝑡) Income for household 𝑖 with state of nature 𝑠 in period 𝑡
𝑢(∙) Utility that household 1 derives from its own consumption
𝑣(∙) Utility that household 2 derives from its own consumption
𝛾 Altruism multiplier
ℎ𝑡 History of states 𝑠 realized before period 𝑡 which dictated previous transfers
𝜏(ℎ𝑡) Transfer function of income from household 1 to household 2
𝑈(ℎ𝑡) Total expected utility for household 1 given history ℎ𝑡
𝑊𝑠(𝑈) Set of implementable contracts which maximize household 2 utility given
household 1 utility 𝑈 for state 𝑠
𝜆(ℎ𝑡) Another formulation for the set of implementable contracts, characterized by
the ratio of marginal utilities between household 1 and household 2
The altruism model starts with the basic model for risk sharing. With altruistic feelings, the utility of a
household comes their own consumption as well as some utility of consumption of other households. In
the case of 2 households with an altruism multiplier 𝛾 < 1, the utility functions are as follows:
Household 1: 𝑢(𝑐1) + 𝛾𝑣(𝑐2)
Household 2: 𝑣(𝑐2) + 𝛾𝑢(𝑐1) (15)
Household 1 & 2 can agree to share risk related to income shocks, but we assume that their informal
contract is not legally enforceable. In this dynamic model, households decide in each period whether to
fulfill their contractual obligation. The risk sharing agreement holds until one household reneges. At that
point, the households move to sequential static Nash Equilibria (SSNE), such that they make their best
decisions non-cooperatively. Because they are still altruistic, they still may give gift & transfers 𝑡 even
without a risk sharing agreement. The transfer function associated with the SSNE are denoted by 𝜏𝑁(𝑠)
which depend on the state of the world 𝑠:
(16)
The function (16) describes the transfer from household 1 to household 2, so the transfer can be positive
or negative. The first line of (16) says: if the ratio of the marginal utility of household 1’s own income to
the marginal utility of household 2’s own income is less than the degree of altruism 𝛾, then household 1
will make a positive transfer to household 2. The second line gives the opposite scenario, where the ratio
is greater than 1/𝛾 and household 1 receives a transfer from 2. Intuitively, the better off household gives
a transfer to the worse off household, unless the ratio is between 𝛾 and 1/𝛾 and no transfer occurs. We
can see that as the degree of 𝛾 increases, this range of 𝛾 to 1/𝛾 increases such that transfers are less likely
in the SSNE scenario.
In each period 𝑡, household 1 makes their decision to oblige or renege the contract based off the utility
differential of the contract with respect to the SSNE scenario. Household 1 takes into account this utility
difference in period 𝑡 plus the discounted differential for all future periods:
The transfer under a risk sharing contract depends on the history ℎ(𝑡) = {𝑠1, 𝑠2, … , 𝑠𝑡−1} of past transfers
for past states of the world. Let 𝑉(ℎ𝑡) be the analogous total utility function for household 2. The risk
sharing contract is enforced if 𝑈(ℎ𝑡) and 𝑉(ℎ𝑡) are both positive so that the contract gives more utility
than reversion to the Nash Equilibrium scenario. Further, a contract is only implementable if 𝑈(ℎ𝑡) ≥ 0
and 𝑉(ℎ𝑡) ≥ 0, so that a time-linked contract is at least as good as a 1 period game from (11).
The set of all optimal contracts (from the set of implementable contracts) maximizes the utility for
household 2, given household 1’s utiliy 𝑈(ℎ𝑡) for a certain state of the world 𝑠 in period 𝑡. The optimal
contract evolves according to marginal utilities of household 1 and household 2 which depend on the
history of transfers ℎ(𝑡):
Difference in utility in current
period between the contracted
transfer and the SSNE transfer
(Discounted) difference in utility
between the contracted transfer
and the SSNE transfer for all
future periods
(17)
The optimal contract in period 𝑡 + 1 depends on the history ℎ(𝑡 + 1). It evolves from the set of
implementable contracts [𝜆𝑠𝐿, 𝜆𝑠
𝑈], along the Pareto Frontier such that:
(18)
In order to analyze the evolution of optimal contracts after random shocks, Foster and Rosenzwieg (2001)
conduct simulations of their altruistic risk sharing model. They specify the form for 𝑢(∙) and 𝑣(∙) and set
parameters for the altruism multiplier 𝛾 and the discount factor 𝛽. They also assume four possible states
of the world with determined income levels for household 1 and household 2; each state has a specified
probability of occurrence. Simulations are conducted for a range of 𝛾 in order to analyze the effect of
altruism on the distribution of risk sharing and total expected utilities. In general, they find that informal
risk sharing arrangements without legal enforcement are limited in efficiency because of imperfect
commitment. They also observe that altruism can increase efficiency and expected gains from risk
sharing.
4.3 Evidence on Motives
Early empirical economic literature on risk sharing focused on altruism and social norms as motives
explaining risk sharing between households. A competing literary has sought to test whether risk sharing
is best explained by altruism or self-interest. Quasi-credits (Platteau and Abraham, 1987) were shown to
arise as a result of commitment constraint. Fafchamps and Lund (2003) find that if there is perfect
enforcement, quasi-credit is unnecessary complication to achieve risk. They also find that mutual
assistance between close relatives take the form of gifts while between distant relatives and friends it take
the form of quasi-credits.
Perfect Enforcement
Perfect enforcement of risk-sharing contract may arise for reasons other than altruism. In the limited
commitment model, participation constraint is not binding when agents are sufficiently patient, high 𝛽.
Perfect enforcement may also be achieved through external agents, since court enforcement is not an
option (insurance contract in question is not explicit), social norms supported by social sanctions is a
more serious enforcing mechanism.
Persistent Shocks
Altruism imposes no restriction on the insurability of persistent shocks, but reciprocal arrangements based
on self-interest are less capable of providing insurance against persistent shocks. De Weerdt and
Fafchamps find no evidence that persistent shocks are less well insured by informal gifts and transfers.
Altruism vs. Social Norms
Gift giving is explained by either altruism or social norms. Then, is gift giving is driven by altruism they
should flow only from rich to the poor, but if they are driven by social norms, they need not. Thus, the
rich receives more gifts. Also, the size of the gift is not driven by differences in wealth or income.
Nature of Transfers
Transfers between households that appear contingent on shocks should be interpreted as evidence of an
intention to share risk. Platteau (1991, 1996) argues that they may ne manifestation of more general
redistribution to unfortunate members of society. The redistribution intention rather than sharing risk is
also supported by the evidence of redistributive social norms and aversion to inequality in rural
Zimbabwean households. Comparing funeral attendance Barr and Stein (2008) found that even though
virtually all households contributed to and attended the funeral, in many cases the head of the household
was absent even more so at the funeral of a wealthy person. It the evidence of disapproval towards
financial success and consistent with existence of distributive norms in the community, because otherwise
the self-interest would dictate a more likely presence at the funeral as the relatives of the deceased are a
source of future insurance.
5. Risk Sharing Groups and Networks
5.1 Groups
Group sharing is the pooling of risk at the level of a group of households. For example, funeral societies,
health insurance groups, self-help groups (SHG), and Rotating Savings and Credit Associations
(ROSCAs) are forms of group sharing.
With larger pooling groups, it is easier to efficiently insure shocks because in smaller risk pools there is a
chance that large shocks overwhelm risk-sharing scheme in any period. With larger groups (e.g. national
health insurance) there are more people to spread the risk. However, there are some restrictions to group
size imposed by voluntary participation in a repeated game setting. Genicot and Ray (2003) is arguably
the best example of this literature. The authors formalize the conditions under which coalitions of
households can credibly oppose larger risk sharing arrangements. By recursion the authors identify the
maximum possible size of self-enforcing risk sharing groups. Their analysis suggests that it is likely to be
small.
Barr and Genicot (2008) found that the nature of enforcement mechanism matters for the size of the risk
sharing pools. External enforcement led to larger risk pooling groups and more risk taking, while intrinsic
enforcement let to smaller groups. When defecting from a group had to be made publicly, risk pooling
was the smallest and people refrain from forming groups with valuable members of the community. Barr
and Genicot also found that assortative matching in gender and age is also used as basis for group self-
selection.
Reasons households opt out of the risk sharing groups
Household have high average income and the equal sharing implies a redistribution of income
from the rich to the poor. If redistribution exceeds the rich’s willingness to pay, their voluntary
participation constraint is violated
Households have identical expected income but different risk aversion. Redistribution is not a
concern. Nearly risk-neutral households would prefer high risk/high gain activities, while highly
risk-averse households will choose “safe” investment. The self-selection is based on the degree of
risk aversion
5.2 Networks
The empirical literature suggests that, in many instances, risk sharing takes place within decentralized and
partially overlapping interpersonal networks. The nodes are agents, and the links are relationships
between them. A star network would be one in which the central node insures all other nodes. The center
household or individual may be a rich “patron” or an insurance company. A polygon would correspond to
a decentralized arrangement in which each node/agent has two helping friends.
Role of Networks
Serve as conduits for costless or low cost transfers, allowing for more efficient insurance. Links
following blood ties and geographic proximity provide for higher mutual gains from risk pooling,
because the transfers are costless and there are fewer commitment constraints
Serve as conduits of information, affecting the severity of punishments for non-compliance with
the scheme; information asymmetry makes risk sharing inefficient
Bloch, Genicot and Ray (2004) take the structure of the interpersonal network as given and identify the
conditions under which decentralized risk pooling between households is the stable equilibrium of a
strategic game. They provide important results relating the shape of the underlying interpersonal network
and the set of sustainable equilibria. It is also suspected that the link formation process itself responds to
strategic considerations. The rapidly emerging literature on social networks provides a wealth of insights
regarding this process.
Link formation
Bilateral: both households agree to form a risk sharing relationship, and it is driven by self-interest and
mutual benefit. An informal insurance network is the collection of bilateral agreements. Only directly
linked agents make transfers to each other and are aware of the aggregate transfers each make to each
other.
Unilateral: unilateral link formation arises if risk sharing is driven by social norms and households cannot
subtract themselves from their obligation to assist others (related to intrinsic motivations like shame and
self-righteousness).
Name of Variables Description
𝑝(𝜃) Probability of a state of nature 𝜃 drawn from a finite set Θ.
𝑦 > 0 Endowment of each agent
𝒚(𝜃)
The vector of income realizations for all agents. Assume symmetry:
if y is the realization at state 𝜃 and 𝑦′ is a permutation of 𝑦, then
there
is another state 𝜃′ with 𝑝(𝜃) = 𝑝(𝜃′) and 𝑦′ = 𝑦(𝜃)
𝑔
Graph, collection of pairs of agents, with the interpretation that the
pair 𝑖𝑗 belongs to 𝑔 if they are directly linked. 𝑁(𝑔) the set of agents
in a graph
𝑑 Component is any maximally connected subgraph of the network.
𝑁(𝑑)
ℵ𝑖(𝑔) ≡ {𝑗|𝑖𝑗} The set of agents directly linked to 𝑖. 𝑧𝑖(𝜃) Total transfers to (or from) third parties
𝑥𝑖𝑗 Transfer, positive or negative, from j to i. Bilateral transfer scheme –
a collection of state-contingent transfers across the two individuals.
𝑥𝑖𝑗(𝜃) = 𝑥(𝑖, 𝑗, 𝑑, 𝑦𝑖 , 𝑦𝑗 , 𝑧𝑖, 𝑧𝑗) Bilateral transfer norm
𝒄𝑖(𝜃) Nonnegative, state-contingent consumption vector
The complete network is just the graph in which all conceivable pairs are directly linked, and the empty
network is the graph in which no pair is linked. A path between agents 𝑖 and 𝑗 is said to exist if there is a
sequence of direct links leading from 𝑖 to 𝑗.
Equal Sharing
A transfer between two agents is chosen to equalize consumption the two agents in each state. In other
words, for each 𝜃, 𝑥𝑖𝑗 solves:
𝑦𝑖(𝜃) + 𝑧𝑖(𝜃) + 𝑥𝑖𝑗(𝜃) = 𝑦𝑗 + 𝑧𝑗(𝜃) − 𝑥𝑖𝑗(𝜃) (19)
Welfare Functions
The transfer norm can be derived from unconstrained welfare maximization 𝑊(𝑖, 𝑗, 𝑑, 𝑐𝑖 , 𝑐𝑗). The transfer
norms can also be derived from constrained welfare maximization. An example is Nash bargaining.
Assume the link 𝑖𝑗 is no longer used. Then 𝑖′𝑠 payoff will depend on the prediction of what will happen in
the ambient network 𝑖𝑗 − 𝑑.
Let 𝑣𝑖(𝑖𝑗 − 𝑑), 𝑣𝑗(𝑖𝑗 − 𝑑) be the expected utilities derived by agents 𝑖 and 𝑗 from the state of affairs.
Then the transfer norm is derived from the Nash product:
[𝐸𝑢(𝑐𝑖(𝜃) − 𝑣𝑖(𝑖𝑗 − 𝑑)] ∗ [𝐸𝑢(𝑐𝑗(𝜃) − 𝑣𝑗(𝑖𝑗 − 𝑑))] (20)
s.t. 𝑉(𝑣𝑖(𝑖𝑗 − 𝑑), 𝑣𝑗(𝑖𝑗 − 𝑑) ≤ 𝑈((𝑢(𝑐𝑖), 𝑢(𝑐𝑗))
Punishment scheme and network stability
Weakest punishment: Only individuals with respect to whom a deviant did not fulfil her obligations break
links with her and the best (from the deviant's viewpoint) stable subnetwork forms. To strengthen
schemes the individuals who are directly connected to the deviant and are 𝑞 links away from the victim,
but not via the deviant, to also break links with the deviant, where q can be made progressively larger to
capture wider information flows. This is called the level 𝑞 – punishment. A network is q-stable when it is
stable for a punishment structure of level-q. The density of links (as well as their specific placement in the
network) can further weaken the punishment and have adverse impact on the network stability.
Strong punishment: A deviating agent is excluded by the entire community, and receives after his
deviation his autarchic allocation. All networks are stable under strong punishment.
There are two conflict ting forces in relation between the architecture of the network and the stability of
insurance schemes. Transit effect, a short-term effect, which comes from the role of links as channel of
transfers, and a information effect that determines the capacity of the network to punish its deviant. These
two aspects play a critical role in the stability of the network. The transit effect is also known as
bottleneck effect - the presence of certain key agents who act as bridges for several transfers.
A bottleneck agent is the agent who has the highest short-term incentive to deviate and the enforcement
constraint faced by this bottleneck agent defines the stability of the entire network. The addition of new
links can only relax the bottleneck effect, as new links can be used to reroute transfers at every state.
Hence, adding links can only improve the stability of the network, and the complete network is stable for
lower values of the discount factor than any other network.
Ellsworth (1989) collected data from one village in Burkina Faso. She found that the local “holy man”
was the “bottleneck agent”. While most of the contributions through this holy man were given to the
community’s disadvantaged, a sizable portion of those ended up in the hands of the man’s brother.
6. Extensions The group risk sharing models described so far include those groups formed with the purpose of
“consumption smoothing” to help members of the group to deal with idiosyncratic risks. Generally, these
risk sharing arrangements assume transfers in the form of gifts or informal credit, quasi-credit, which
have a purpose to deal with different kinds of shocks resulting in interruption of earnings of the group’s
members, (for example, loss in earnings due to temporary disability). Thus, this model is not designed to
reflect the risk pooling groups, which have a purpose of “cash flow” smoothing. Notice that, even quasi-
credit are not used for investments. Such particular groups are called ROSCAs, which primarily serve the
needs of entrepreneurs in the developing countries.
In ROSCAs each member of the group contributes an equal sum of money and members agree to meet a
specified number of times per period.
Name of variable Description
𝑖 ∈ 𝑁 Number of participants in a group
𝑘 Number of “turns” in a cycle. N=K, i.e. a every member should get
the “pot” exactly 1 time during the cycle
𝑑 Duration of the cycle, any length of period agreed on
𝑡𝑘 Time is now measured by the length of time between each meeting
𝑡𝑘 − 𝑡𝑘−1 The length of time between each turn
𝑐𝑡𝑘
𝑖 (𝜏𝑎) Consumption is now a function of the aggregate sum of transfers, 𝜏𝑎 ,
a member of the group will receive on his/her turn, since we now
assume that the transfers received are reinvested. 𝜏𝑎 = ∑ 𝜏𝑖𝑘𝑖=1
𝐴𝑇𝑖 > 0
Non-strategic penalty for defection: Subjective satisfaction from
being part of the group; guilt; social exclusion. The idea is partially
borrowed from Fafchamps (1999)
A rotation order is determined before the first transfer is made. Therefore, it will be interesting to test
whether the order of the member in the rotation cycle will matter in maximizing his/her utility from
participating in the risk-sharing mechanism. For example, as a result of a lottery agent 𝑖 is placed the last,
then the RHS of the voluntary participation constraint below will be lower than the RHS of (10).
𝑈𝑖(𝑦𝑠𝑘𝑖 ) − 𝑈𝑖(𝑐𝑠𝑘
𝑖 ) ≤ ∑ 𝛽𝑡𝑘𝐸𝑠𝑘(𝑈𝑖 (𝑐𝑡
𝑖(𝜏𝑎)) − 𝑈𝑖(𝑦𝑡𝑖)𝑇→∞
𝑡𝑘=𝑠𝑘+1 ) (21)
Also, what if 𝜏𝑎 < ∑ 𝜏𝑖𝑘𝑖=1 , i.e. some of the members of the of the group renege on their duties before the
group and do not make contributions. How would that affect the decision of the agent 𝑖 to continue
participate in ROSCA.
Finally, Threat of exclusion, Information sharing, guilt and shame are recognized as informal risk-sharing
contract enforcement tools. We can include a variable 𝐴𝑇𝑖 > 0, captures all the “moral” consequence of
defection from the group, but we make it as a function of T, which is the length of time agent 𝑖 has spent
in the group. This way we reflect the fact that the consequence of reneging on one’s duties are not only
affected by the norms existent in the community as a whole, but also exacerbated by the “bonding” that
among members that has taken place. Therefore,
𝑈𝑖(𝑦𝑠𝑘𝑖 ) − 𝑈𝑖(𝑐𝑠𝑘
𝑖 ) ≤ ∑ 𝛽𝑡𝑘𝐸𝑠𝑘(𝑈𝑖 (𝑐𝑡
𝑖(𝜏𝑎)) − 𝑈𝑖(𝑦𝑡𝑖)𝑇→∞
𝑡𝑘=𝑠𝑘+1 + 𝐴𝑇𝑖 (22)
This condition for voluntary participate provides for much more stable and efficient risk sharing.
7. References Barr, A. and G. Genicot (2008). Risk Sharing, Commitment and Information: An Experimental Analysis.
Journal of Development Economics 68(2): 355-379.
Bloch, F., G. Genicot, and D. Ray. 2004. Social Networks and Informal Insurance. (mimeograph).
Fafchamps, Marcel. 2008. Risk Sharing Between Households. Chapter for Handbook of Social
Economics, eds. Jess Benhabib, Alberto Bisin, and Matthew O. Jackson.
Fafchamps, Marcel. 1999. Rural Poverty, Risk, and Development. Report for Food and Agricultural
Organization.
Foster, A.D. and M.R. Rosenzweig. 2001. Imperfect Commitment, Altruism, and the Family: Evidence
from Transfer Behavior in Low-Income Rural Areas. Review of Economics and Statistics: 83(3): 389-407.
Freeman, Linton. 2000. Visualizing social networks. Journal of Social Structure 1(1).
Genicot, G. and D. Ray. 2003. Group Formation in Risk-Sharing Arrangements. Review of Economics
Studies 70(1): 87-113.
Platteau, J.P. and Abraham, A. 1987. An Inquiry in Quasi-Credit Contract: The Role of Reciprocal Credit
and Interlinked Deals in Small-Scale Fishing Communities. J Dev. Stud. 23(4): 461-490.
Rosenzweig, M.R. and O. Stark. 1989. Consumption Smoothing, Migration, and Marriage. Journal of
Political Economy 97(4): 905-26.
