Natural disaster and risk-sharing behavior: Evidence from rural Bangladesh

Abstract

Using a unique field experiment in rural Bangladesh, this paper investigates how exposure to a natural disaster affects risk-sharing behavior. We conducted a risk-sharing experiment that randomly assigned different levels of risk-sharing commitments to individuals who were exposed and unexposed to a recent natural disaster and asked them to form risk-sharing groups. Our results show that disaster-affected individuals are less likely to defect from risk-sharing groups, regardless of the level of ex-ante commitment. Interestingly, individuals from disaster-affected villages chose riskier bets and realized higher average returns compared with individuals from non-disaster-affected areas. Our results have important implications for the design of financial risk-transfer mechanisms in developing countries.

Appendix

1.1 A model of risk-sharing and defection

Here, we develop a model of risk-sharing and defection. For simplicity, we assume a common defection cost c ≥ 0, which may depend on whether a natural disaster previously occurred in the region, the public visibility of defection, and prevailing social norms. We let $u_{i} \left (x \vert \gamma _{i} \right )$ denote agent i’s vNM utility function over certain monetary payments x > 0, for γ_i > 0 agent i’s risk aversion parameter. $u_{i} \left (x \vert \gamma _{i} \right )$is assumed to be increasing and strictly concave in x, and:

$$ \frac{\partial }{\partial \gamma_{i}}\frac{\frac{\partial^{2}}{\partial x^{2}}u_{i} \left( x \vert \gamma_{i} \right) }{\frac{\partial }{\partial x}u_{i} \left( x \vert \gamma_{i} \right) }<0. $$

We also assume that $\frac {\partial }{\partial \gamma _{i}}u_{i} \left (x \vert \gamma _{i} \right ) \leq 0$, $\frac {\partial }{\partial \gamma _{i}}\frac {\partial }{\partial x}u_{i} \left (x \vert \gamma _{i} \right ) \leq 0$ and $\frac {\partial }{\partial \gamma _{i}}\frac {\partial ^{2}}{\partial x^{2}}u_{i} \left (x \vert \gamma _{i} \right ) \leq 0$ when x is non-negative. That is, the utility agents receive from cash payments is decreasing as they become more risk-averse. These properties are satisfied for both constant relative risk aversion (CRRA):

$$ u_{i} \left( x \vert \gamma_{i} \right) = \left\{ \begin{array}{ll} \frac{x^{1- \gamma_{i}}-1}{1- \gamma_{i}} & if \gamma_{i} \neq 1\\ ln \left( x \right) & if \gamma_{i}=1 \end{array}\right. $$

and constant absolute risk aversion (CARA):

$$ u_{i} \left( x \vert \gamma_{i} \right) =\frac{1-e^{- \gamma_{i}x}}{ \gamma_{i}} $$

Players choose (i) whether to share risk by joining a pool, (ii) a lottery $\tilde {x}_{i}$, and when joining a pool (iii), a defection strategy d_i, which we describe as follows: if player i joins a pool P (a set of pool members), their expected payoff is:

$$ {V_{i}^{P}} \left( \tilde{x},d \right) =E_{\tilde{x}} \left[ u_{i} \left( d_{i} \left( \tilde{x}_{i} \right)\tilde{x}_{i}+ \left( 1-d_{i} \left( \tilde{x}_{i} \right) \right) \tilde{y} \vert \gamma_{i} \right) \right] -E_{\tilde{x}_{i}} \left[ d_{i} \left( \tilde{x}_{i} \right) \right] c $$

where $\tilde {x} \equiv \left (\tilde {x}_{j} \right )_{j \in P}$ denotes the profile of lotteries that players in P choose, we denote:

$$ y=\frac{ {\sum}_{j \in P}^{}d_{j} \left( x_{j} \right) x_{j}}{ {\sum}_{j \in P}^{}d_{j} \left( x_{j} \right) }, $$

equal to the total payment to i when in the pool, $d_{i} \left (x_{i} \right ) \in \left \{ 0,1 \right \}$ gives i’s lottery-outcome-contingent defection strategy (i.e., $d_{i} \left (x_{i} \right ) =1$ iff defect), and we denote the profile of defection strategies by $d \equiv \left (d_{j} \right )_{j \in P}$. Each player’s action is the choice $\tilde {x}_{i} \in X$, where X gives the set of lotteries to choose from and the defection strategy $d_{i} \left (x \right )$, which is either “no defection” or “defection” depending on the outcome of $\tilde {x}_{i}$ (i.e. x_i). We assume that X is ordered, with a “greater” $\tilde {x}_{i}$ yielding a greater expected return and greater variance (riskiness). To simplify the environment, we focus on the experimental design where lotteries are determined via a fair coin toss. That is, $\tilde {x}_{i}$ is x_i,HIGH and x_i,LOW with probability 1/2 each.¹³ When convenient, we assume that the set X is smooth, with $x_{i,HIGH} \left (\tilde {x}_{i} \right )$ and $x_{i,LOW} \left (\tilde {x}_{i} \right )$ continuous and differentiable in the chosen lottery $\tilde {x}_{i}$ (note that the implemented design takes finite |X| = 6).

If player i does not join a pool, their expected payoff is simply:

$$ V_{i}^{NP} \left( \tilde{x}_{i} \right) =E_{\tilde{x}_{i}} \left[ u_{i} \left( \tilde{x}_{i} \vert \gamma_{i} \right) \right] . $$

Thus, i will choose between the optimal strategy d maximizing ${V_{i}^{P}} \left (\tilde {x}^{P},d \right )$ versus not joining the pool which yields $V_{i}^{NP} \left (\tilde {x}_{i}^{NP} \right )$, where $\tilde {x}^{P}$ gives the profile of optimal lotteries when entering the pool and $\tilde {x}_{i}^{NP}$ gives i’s optimal lottery when not entering the pool.

Whether agent i joins a pool or not, their choice of $\tilde {x}_{i}$ will determine the expected return and riskiness associated with the lottery. If this agent does not join a pool, this lottery is simply $\tilde {x}_{i}=\tilde {x}_{i}^{NP}$. If the agent does join a pool, this lottery is $\tilde {y}$ under $\tilde {x}_{i}=\tilde {x}_{i}^{P}$. In either case, the expected return and riskiness of the lottery that i faces is increasing in the expected return and riskiness of $\tilde {x}_{i}$. Under our assumptions on $u_{i} \left (x | \gamma _{i} \right )$ and X, the following can be shown:

Proposition 1

The expected return and riskiness of optimal $\tilde {x}_{i}$ in X is decreasing in γ_i.

Proof

Online Appendix A. □

Observation 1

Proposition 1 is consistent with the conclusion that individuals directly exposed to a disaster become more risk tolerant in order to secure increased gains.

We turn to defection incentives when agents join a risk-sharing pool. If i enters the pool and takes a defection strategy $d_{i} \left (x_{i} \right ) =1$ if x_i = x_i,HIGH and $d_{i} \left (x_{i} \right ) =0$ if x_i = x_iLOW, then $V_{i} \left (\tilde {x},d \right )$ can be written as follows:

$$ V_{i} \left( \tilde{x},d \right) =V_{i}^{P-D} \left( \tilde{x},d_{-i} \right) \equiv \frac{1}{2} \left( u_{i} \left( x_{i,HIGH} \vert \gamma_{i} \right) -c \right) +\frac{1}{2}E_{\tilde{x}_{-i}} \left[ u_{i} \left( y_{i,LOW} \vert \gamma_{i} \right) \right] , $$

where:

$$ y_{i,LOW}=\frac{x_{i,LOW}+ {\sum}_{j \in P}^{}d_{j} \left( x_{j} \right) x_{j}}{1+ {\sum}_{j \in P^{}d_{j}} \left( x_{j} \right) }. $$

If i takes a no-defection strategy ($d_{i} \left (x_{i} \right ) =0$ always), then:

$$ V_{i} \left( \tilde{x},d \right) =V_{i}^{P-ND} \left( \tilde{x},d_{-i} \right) \equiv \frac{1}{2}E_{x_{-i}} \left[ u_{i} \left( y_{i,HIGH} \vert \gamma_{i} \right) \right] +\frac{1}{2}E_{x_{-i}} \left[ u_{i} \left( y_{i,LOW} \vert \gamma_{i} \right) \right] , $$

where:

$$ y_{i,HIGH}=\frac{x_{i,HIGH}+ {\sum}_{j \in P}^{}d_{j} \left( x_{j} \right) x_{j}}{1+ {\sum}_{j \in P}^{}d_{j} \left( x_{j} \right) }. $$

Clearly, defection is preferred to no defection iff $V_{i}^{P-D} \left (\tilde {x},d_{-i} \right ) >V_{i}^{P-ND} \left (\tilde {x},d_{-i} \right )$, equivalently:

$$ u_{i} \left( \tilde{x}_{i,HIGH} \vert \gamma_{i} \right) -E_{\tilde{x}_{-i}} \left[ u_{i} \left( y_{i,HIGH} \vert \gamma_{i} \right) \right] >c. $$

This provides our next result.

Proposition 2

The incentives to defect in a pool depend on the pool size, the lotteries and defection strategies of others in the pool, the cost of defection c, and on the size of the high outcome of the lottery chosen by the defecting individual; incentives to defect are independent of the low outcome of the lottery chosen by the defecting individual.

Now, further assume that u_i = u and γ_i = γ for all i (γ may still depend on the treatment). Then, by the symmetry of the setting, a symmetric-pool equilibrium can be obtained where each member of the pool takes a similar strategy, both in their chosen lottery and defection strategies. Indeed, when a symmetric-pool equilibrium is obtained all pool members choose a lottery $\tilde {x}^{P\ast }$, and the value to defection upon realizing $x_{i,HIGH}^{P\ast }$, equal to $u \left (x_{i,HIGH}^{P\ast } \vert \gamma \right ) -E_{\tilde {x}_{-i}} \left [ u \left (\tilde {y}_{i,HIGH}^{P\ast } \vert \gamma \right ) \right ]$, is always positive. When all others do not defect, then $\tilde {y}_{i,HIGH}^{P\ast }=\tilde {y}_{i,HIGH}^{ND\ast } \equiv \left (x_{i,HIGH}^{P\ast }+ {\sum }_{j \in P}\tilde {x}_{j}^{P\ast }\right )/n$, for n = |P| giving the size of the pool. Alternatively, when all others choose to defect, the incentives to enter and contribute to the pool (by not defecting) dissolve. Therefore, the value to defection ${\Delta }^{D\ast } \equiv u \left (x_{i,HIGH}^{P\ast } \vert \gamma \right ) -E_{x_{-i}} \left [ u \left (y_{i,HIGH}^{ND\ast } \vert \gamma \right ) \right ]$ when all others do not defect pins the threshold such that no members defect in equilibrium if c ≥Δ^D∗, and the pool disbands if c < Δ^D∗. Indeed, Δ^D∗ defines the equilibrium propensity to defect, which we describe next.

1.2 A.1 Comparative statics

Given that many variables are changing upon moving between the control and treatment (the risk aversion parameter γ, the optimal $x_{i}^{NP}$ and x^P∗, the defection cost c, and the size of the pool n ), here we will assess the comparative statics of a symmetric-pool equilibrium with respect to each variable, ceteris paribus(i.e., we provide a partial-equilibrium analysis). The broad takeaway will be that an increase in the defection cost c is necessary to rationalize an observed decrease in defection in disaster regions.

1.2.1 A.1.1 Risk-sharing and defection

For our first set of comparative statics results, we focus on defection behaviour under risk-sharing (i.e., joining a pool), and set aside the endogenous $\tilde {x}_{i}^{NP}$ and the corresponding choice between joining and not joining a pool. We then consider the incentives to join the pool as risk preferences change in the following subsection.

First, we hold the endogenous $\tilde {x}^{P\ast }$ and n fixed and marginally increasing γ (note that the former will hold for local changes to γ).

Proposition 3

In the symmetric-pool equilibrium, the cut-off Δ^D∗ is decreasing in the coefficient of risk aversion γ.

Proof

Online Appendix A. □

In words, holding all else equal, the incidence of defection (i.e., the range of c < Δ^D∗) decreases as the agents become more risk-averse. The reason is as follows: as agents become more risk averse, the gain to absconding with the high payment $x_{i,HIGH}^{P\ast }$ decreases. Conversely, the opportunity cost of remaining in the pool decreases less abruptly, and approaches the value to defection. In total, the defection gain Δ^D∗ decreases.

Observation 2

With subjects exhibiting significantly less risk aversion in the disaster treatment (revealed in the first task), the finding that less defection is observed in the disaster treatment is inconsistent with Proposition 3.

Second, we hold γ and n fixed and increase the risk of endogenous $\tilde {x}^{P\ast }$ chosen from X. With an increase to the mean and standard deviation of $\tilde {x}^{P\ast }$, the effect on $E_{\tilde {x}_{-i}} \left [ u_{i} \left (\tilde {y}_{i,HIGH}^{ND\ast } \vert \gamma \right ) \right ]$ is ambiguous. Therefore, for the following result, we impose an additional structure to the form of $u \left (x \vert \gamma \right )$ and to the ordering of X. First, we assume that $u \left (x \vert 0 \right ) =a+bx$ for some b > 0 (as with CRRA utility). Second, we take lotteries $\tilde {x}$ and $\tilde {x}^{\prime }$ such that $\tilde {x}- \delta$ gives a mean-preserving spread of $\tilde {x}^{\prime }$ for some δ ≥ 0. In the experimental design, δ = 40 Taka between lottery options 1 through 5 (δ = 0 between options 5 and 6). We say that $\tilde {x}$ and $\tilde {x}^{\prime }$ are ordered “simply” if $\tilde {x}$ can be obtained from $\tilde {x}^{\prime }$ via such homogenous upward shifts coupled with mean-preserving spreads. For the next result, which establishes sufficient conditions for Δ^D∗ to be increasing in the riskiness of $\tilde {x}^{P\ast }$, we assume the lotteries $\tilde {x}^{P\ast } \in X$ are ordered “simply”.

Proposition 4

In the symmetric-pool equilibrium, there is some γ > 0 and δ > 0 such that for all γ < γ and δ < δ, the cut-off Δ^D∗ is increasing in the riskiness of $\tilde {x}^{P\ast }$.

Holding all else equal, the incidence of defection (i.e., the range of c < Δ^D∗) increases as the agents bring greater risk to the risk-sharing pool, provided that the agents are not too risk averse and the additional expected value of each lottery is not too large. The intuition is as follows: as agents bring more risk to the pool, the gain from absconding with the high payment $x_{i,HIGH}^{P\ast }$ increases. Conversely, the opportunity cost of remaining in the pool increases more slowly, provided the agents are not too risk-averse and the additional expected return is not too large. Consequently, greater x^P∗ under small γ and δ yields an increasing value to defection.

We exhibit the bounds of the result numerically, as follows: Table 8 provides Δ^D∗ values for various γ values under CRRA utlity. For γ values of 0.5 and 1, Δ^D∗ increases monotonically across the lottery choices of the experimental design; note that Δ^D∗ = 0 for lottery 1. For γ = 2, this trend reverses between lottery 4 and 5, and for γ = 4, the trend reverses across lotteries 1 to 5. For lottery 6, which gives a mean-preserving spread of lottery 5, Δ^D∗ always increases.

Table 8 Lottery Table 1

In Table 9 , 1000 Taka is added to all payoffs, capturing a non-zero initial level of wealth. Now, the Δ^D∗ increases monotonically across all lottery choices. We see that the domain of Proposition 4 is quite broad, particularly when including non-zero initial wealth.

Table 9 Lottery Table 2

Observation 3

With subjects taking significantly greater risk to the pool in the disaster treatment, the findings that agents are less risk-averse and less defection in the disaster treatment are inconsistent with Proposition 4.

Next, we hold γ and x^P∗ fixed and increase the size of the pool n. Now, the risk associated with staying in the pool decreases (because of the diversification in $y_{i,HIGH}^{ND \ast }$ ), while the expected return remains fixed. It is straight forward to show the following:

Proposition 5

In the symmetric-pool equilibrium, the cut-off Δ^D∗ is decreasing in n.

Proof

Online Appendix A. □

Observation 4

With the average size of the pools in the disaster treatment significantly smaller than those of the control, the finding that less defection is observed in the disaster treatment is inconsistent with Proposition 5.

Main Observation

Given that Propositions 3, 4 and, 5 are each inconsistent with the experimental findings, a significant increase in the defection cost c is necessary to rationalize the observation that defection decreases significantly in the disaster treatment.

1.2.2 A.1.2 Risk-sharing choice

The comparative statics for the values of agents choosing not to pool are straight-forward. Precisely, given that the expected return and variance options from lottery choices in set X remain the same, facing less aversion to risk implies that agent i enjoys greater expected utility, both directly through a decrease in γ and indirectly through the chosen riskier $\tilde {x}_{i}^{NP}$ (under our assumptions on $u_{i} \left (x \vert \gamma \right )$). That is, each i faces less cost when it comes to bearing risk and reaps more gains when it comes to greater expected return. The comparative statics on values to agents choosing to share risk, however, are less straight-forward given that the expected utilities depend on the risk choices of other pool members. For simplicity, we consider the comparative static holding fixed $\tilde {x}=\tilde {x}^{NP}$ = $\tilde {x}^{P\ast }$. Moreover, we consider the case of c > Δ^D∗, where the pool can exist. The following is immediate:

Proposition 6

In the symmetric-pool equilibrium, the gain to sharing risk via joining the pool without defection, $V_{i}^{P-ND} \left (\tilde {x},d_{-i}^{ND} \right ) -V_{i}^{NP} \left (\tilde {x} \right )$, is non-decreasing in both γ and n.

Intuitively, incentives to share risk are increasing with risk aversion and the size of the pool.

Observation 5

With subjects exhibiting significantly less risk aversion in the disaster treatment (revealed in the first task), the finding that the frequency of subjects joining a pool is significantly lower in the disaster treatment is consistent with Prediction 5.

Note that the observed lower frequency of joining pools and the significantly lower pool size in the disaster treatment group are complementary observations.